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THE 


Nature  and  Utility 


OF 


MATHEMATICS, 


WITH  THE   BEST   METHODS   OF  INSTRUCTION  EXPLAINED 
AND  ILLUSTRATED 


BY 

CIIAKLES  DA  VIES,  LL.D., 

EMERITUS  PKOFESSOR  OP  HIGHER  MATHEMATICS   IN  COLUMBIA  COLLEOB. 


loavoH  cQii^^^  ]:-^^' 

CHESTNUT  HUX,  MA. 

NEW  YORK: 
PUBLISHED   BY  A.  S.  BARNES  &  CO., 

Ill  AND  113  William  Street. 
1873. 


BAVIES'  MATHEMATICS. 


IN    THRKE    PARTS. 


I-GOMMOIT   SCHOOL  COURSE. 

©avies'  Primary  Aritliiiietic— The  fundamental  principles  clisplaycd  in 
Object  Lessons. 

Eavios'  Iiitellectsial  Aritliiiietic— Eereniii,<j  all  operations  to  the 
luut  1  as  the  only  tangible  basis  for  logical  development. 

Davie*'  Elements  ol"  \yritten  Aritlimetic.— A  practical  inlroductioii 
lo  the  wliolc  subject.     Tlieojy  subordinated  to  Practice. 

I5avic«i'  IPraotieal  Aritlimetic— The  combination  ofTheory  and  Practice, 
inieuded  to  be  clear,  exact,  brief,  and  comprehensive. 

II.-ACADEMrG  COUESE. 

Davies'  University  Aritlimetic— Treating  the  subject  exhaustively  as 
a  6ci«?ice,  in  a  logical  series  of  connected  propositions. 

Daviesi'  "Elementary  Algebra,— A  connecting  link,  conducting  the  pupil 
t-asily  from  arithmetical  processes  to  abstract  analysis. 

BJavles' University  Algebra. — For  institutions  desiring  a  more  complete 
but  not  the  fullest  course  in  pure   Algebra. 

Bavies'  Practical  IHatliematics.— The  science  practically  applied  to  the 
useful  arts,  as  Drawing,  Architecture,  Surveying,  Mechanics,  etc. 

©avies'  Elementary  Geometry.— The  important  principles  in  simple 
lor.n,  but  with  all  the  exactness  of  rigorous  reasoning. 

©avics'  Elements  of  Surveying.— Re-\vritten  in  18T0.  A  simple  and 
prj,ctical  presentatiou  of  the  suljject  for  the  scholar  and  surveyor. 

IIL-GOLLEGIATE  COUESE. 

Davies'  Bourdon's  Algebra.— Embracing  Sturm's  Theorem,  and  a  most 
exhaustive  and  scholarly  course. 

Wavles'  University  Algebra.— A  shorter  course  than  Bourdon,  for  Insti- 
tutions having  less  lime  to  give  the  subject. 

Bavies'  ILegendre's  fieonietry.— The  original  is  the  best  Geometry  of 
Europe.    The  revised  edition  is  well  known. 

Davics'  Analytical  CJeometry. —Being  a  full  course,  embracing  the 
equation  of  surfaces  of  the  second  degree. 

Davies'  Bifflerential  and  Integral  Calculns.— Constructed  on  the 
basis  ot  Coutinuous  Quantiiy  and  Consecutive  Diflercnces. 

Oavies'  Analytical  Geometry  and  Calculus.— The  shorter  treatises, 
combined  in  one  volume,  as  more  available  for  American  courses  of  study. 

BJavies'  Bescriptive  Geometry.— With  application  to  Spherical  Trigo- 
nometry, Spherical  Pi  ejections,  and  Warped  Surfaces. 

Davies'  Shades,  Sliadotvs,  and  Perspective.— A  succinct  exposition 
of  the  mathematical  principles  involved. 

Davies  &  Peck's  JWatliematical  Dictionary.— Embracing  tlie  defini- 
tions of  all  the  terms,  and  also  a  Cyclopedia  of  Mathematics. 

Davies'  Nature  and  Utility  of  IWatliematics.— Embracing  a  con- 
densed Logical  Analysis  of  the  entire  Science,  and  of  its  General  Uses. 


Entered  according  to  Act  of  Coiiprc.ss.  in  the  yc.ir  Eigliteen  Hundred  and  Seventy-three, by 

CHARLES    DAVIES, 

In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


i?4359 


PREFACE. 


The  following  work  is  not  a  series  of  speculations.  It  is 
but  an  analysis  of  that  system  of  mathematical  instruction 
which  has  been  steadily  pursued  at  the  Military  Academy 
nearly  half  a  century,  and  which  has  given  to  that  institu- 
tion its  celebrity  as  a  school  of  mathematical  science. 

It  is  of  the  essence  of  that  system  that  a  principle  be 
taught  before  it  is  applied  to  practice;  that  general  princi- 
ples and  general  laws  be  taught,  for  their  contemplation  is 
far  more  improving  to  the  mind  than  the  examination  of 
isolated  propositions;  and  that  when  such  jorinciples  and 
such  laws  are  fully  comprehended,  their  applications  be  then 
taught,  as  consequences,  or  practical  results. 

This  view  of  education  led,  at  an  early  day,  to  the  union 
of  the  French  and  English  systems  of  Mathematics.  By 
this  union  the  exact  and  beautiful  methods  of  generaliza- 
tion, which  distinguish  the  French  school,  were  blended 
with  the  practical  methods  of  the  English  system.  • 

The  fruits  of  this  new  system  of  instruction  have  been 
abundant.  The  graduates  of  the  Military  Academy  have 
been  sought  for  wherever  science  of  the  highest  grade  has 
been  needed.  Russia  has  sought  them  to  construct  her, 
railroads;*  the  Coast  Survey  needed  their  aid;  the  works  of 
internal  improvement  of  the  first  class  in  our  country,  have 
mostly  been  conducted  under  their  direction ;  and  the  war 
with  Mexico  afforded  ample  oppor. unity  for  showing  the 
thousand  ways  in  which  science — the  highest  class  of  knoAvl- 
edge — may  be  made  available  in  practice. 

®  Major  Whistler,  the  engineer,  to  whom  was  intrusted  the  great  en- 
terprise of  constructing  a  railroad  from  St.  Petersburg  to  Moscow,  and 
Major  Brown,  who  succeeded  him  at  his  death,  were  both  graduates  of 
the  Military  Academy. 


PREFACE, 


All  these  results  are  due  to  the  system  of  instruction.  In 
that  system,  Mathematics  is  the  basis — Science  precedes  Art 
— Theory  goes  before  Practice — the  general  formula  em- 
braces all  the  particulars. 

Although  my  official  connection  with  the  Military  Aca- 
demy was  terminated  many  years  since,  yet  the  general 
system  of  Mathematical  instruction  has  not  been  changed. 
Younger  and  able  professors  have  extended  and  developed 
it,  and  it  now  forms  an  important  element  in  the  education 
of  the  country. 

The  present  work  is  a  modification,  in  many  important 
particulars,  of  the  Logic  and  Utility  of  Mathematics,  pub- 
lished in  the  year  1850.  The  changes  in  the  Text,  seemed 
to  require  a  change  in  the  Title. 

It  was  cfeemed  necessary  to  the  full  development  of  the 
plan  of  the  work,  to  give  a  general  view  of  the  subject  of 
Logic.  The  materials  of  Book  I.  have  been  drawn,  mainly, 
from  the  works  of  Archbishop  Whately  and  Mr.  Mill.  Al- 
though the  general  outline  of  the  subject  has  biit  little  re- 
semblance to  the  work  of  either  author,  yet  very  much  has 
been  taken  from  both ;  and  in  all  cases  Avhere  it  could  be 
done  consistently  with  my  own  plan,  I  have  adopted  their 
exact  language.  This  remark  is  particularly  applicable  to 
Chapter  III.,  Book  I.,  which  is  taken,  with  few  alterations, 
from  Whately. 

For  a  full  account  of  the  objects  and  plan  of  the  work,  the 
reader  is  referred  to  the  Introduction. 


FisHKiLL  Landing,   } 
January,  1873.       J 


CONTENTS 


INTRODUCTION. 

FAGS 

Objects  and  Plan  op  the  Wokk.  .......................  11 — 37 


B  0  0  K    L 

LOGIC. 

CHAPTER  1. 

Definitions — Operations  of  the  Mind — Terms  defined.  27 — 41 


SECTION 

Definitions 1 —  6 

Operations  of  the  Mind  concerned  in  Reasoning 6 — 13 

Abstraction 13  —14 

Generalization 14 — 33 

Terms — Singular  Terms — Common  Terms 15 

Classification. 16—30 

Nature  of  Common  Terms 30 

Science 21 

Art.... 23 


CONTENTS. 


CHAPTER  II. 

PAGE 

SotmcES  AND  Means  op  Knowledge — Indtjction 41 — 54 


SECTION 

Knowledge 23 

Facts  and  Truths 24—27 

Intuitive  Truths 27 

Logical  Truths 28 

Logic 29 

Induction 30 — 34 


CHAPTER  in. 

Deduction — Natuke  of  the  Syllogism: — Its  Uses  and  Ap-      paob 
plications 54 — 97 

SECTION 

Deduction 34 

Propositions 35 — 40 

Syllogism 40—42 

Analytical  Outline  of  Deduction 42 — 67 

Aristotle's  Dictum 54 — 61 

Distribution  and  Non-distribution  of  Terms .....; 61 — 67 

Rules  for  examining  Syllogisms 67 

Of  Fallacies 68—71 

Concluding  Remarks 71 — 75 


CONTEIS'TS. 


BOOK    II. 

MATHEMATICAL   SCIENCE. 

CHAPTER  I. 

Quantity  and  Mathematical  Science  defined — Differ- 
ent KINDS  OP  Quantity  —  Language  of  Mathematics 
EXPLAINED  —  Subjects  Classified  —  Unit  of  Measure 
defined — Mathematics  a  Deductive  Science page 


SECTIOn 

Quantity — Defined : . . .  75 

Mathematics  Defined 76 

Kinds  of  Quantity — Number  and  Space 77 —  87 

Language  of  Mathematics 87 —  91 

Language  of  Number — Geometry — Analysis 91 —  98 

Pure  Mathematics 98 — 104 

Mixed  Matliematics 104 — 105 

Quantity  Measured 105 — 108 

Comparison  of  Quantities 108 — 109 

Axioms — Equality — Inequality 109 — 111 


CHAPTER  II. 

PAGB 

Arithmetic — Science  and  Art  op  Numbers 119 

SECTION    L 

SECTIOH 

First  Notions  of  Numbers Ill — 114 

Ideas  of  Numbers  Generalized 114 — 117 

Unity  and  a  Unit  Defined 117 

Simple  and  Denominate  Numbers 118 — 120 

Alphabet — Words — Grammar 120 

Aritlimetical  Alphabet 121 

Spelling  and  Reading  in  Addition 122 — 127 

Spelling  and  Reading  in  Subtraction 127 — 129 


CONTENTS. 


SECTION 

Spelling  and  Reading  in  Multiplication 129 

Spelling  and  Reading  in  Division 130 

Units  increasing  by  the  Scale  of  Tens 131 — 138 

Units  increasing  by  Varying  Scales 138 

Integral  Units  of  Arithmetic 139 — 141 

Different  Kinds  of  Units 141—157 

Advantages  of  the  System  of  Units 157 — 158 

Metric  System 158—159 

System  of  Unities  applied  to  the  Four  Ground  Rules  .  159 — 163 


SECTION   II. 

Fractional  Units  changing  by  the  Scale  of  Tens 163 — 166 

Fractional  Units  in  general 166 — 169 

Advantages  of  the  System  of  Fractional  Units 169 — 171 


SECTION    III. 
Proportion  and  Ratio 171 — 180 

SECTION    IV. 
Applications  of  the  Science  of  Arithmetic. ...........  180 — 188 

SECTION    V. 

Methods  of  teaching  Arithmetic  considered  ..........  188 

Order  of  Subjects 188—190 

Abstract  Units 190—193 

Fractional  Units 192—193 

Denominate  Units 193—194 

Ratio  and  Proportion 194 — 196 

^A^rithmetical  Language 196 — 204 

Necessity  of  Exact  Definitions  and  Terms 204 — 210 

How  should  the  Subject  be  presented 210 — 213 

Text-Books 213—218 

First  Arithmetic 218—231 

Second  Arithmetic 231 — 235 

Third  Arithmetic 235—240 

Concluding  Remarks. 240—241 


COIvrTENTS. 


CHAPTER   III. 

Geometry  defined — Things  of  which  it  treats — Com:- 
PAnisoN  AND  Properties  op  Figures — Demonstration 
Proportion — Suggestions  for  Teaching page  219 

SECTION 

Geometry 241 

Tilings  of  which  it  treats 242—253 

Comparison  of  Figures  witli  Units  of  Measure 253 — 260 

Properties  of  Figures .  260 

Marks  of  what  may  be  proved 261 

Demonstrations 263 — 271 

Proportion  of  Figures 271 — 274 

Comparison  of  Figures 274 — 277 

Recapitulation — Suggestions  for  Teachers 277 


CHAPTER  IV. 

Analysis  —  Algebra — Analytical  Geometry — Differ-         page 
ential  and  Integral  Calculus 257 


SECTION 

Analysis 278—284 

Algebra 284 

Analytical  Geometry 285—287 

Differential  and  Integral  Calculus 287—290 

Algebra  further  considered 290 — 300 

Minus  Sign 300 — ^302 

Subtraction 302 

Multiplication 303—306 

Zero  and  Infinity 306 — 311 

Of  the  Equation 311—315 

Axioms 315 

Equality — its  Meaning  in  Geometry 316 

Suggestions  for  those  who  teach  Algebra -  319 

1* 


10  COlsrTEXTS. 


CHAPTER  V. 

PAGE 

DiPFERENTIAI,  CaLCTOUS 289 


SECTION 

Foundations  of  Mathematical  Science 320 — 332 

Limits — of  Discontinuous  Quantity 332 — 325 

Given  Quantity — Continuous  Quantity 324 — 326 

Consecutive  Quantities  and  Tangents ...    326 — 329 

Lemmas  of  Newton 329 — 337 

Fruits  of  Newton's  Theory 337—342 

Different  Definitions  of  Limits 342 — 343 

What  Quantities  are  denoted  by  0 343 

Inscribed  and  Circumscribed  Polygons  ...    344 

Differential  Calculus — Language 344 — 346 


APPENDIX. 

PAGE 

A  CoTJESE  OF  Mathematics — What  it  shotild  be 837 


BOOK   III. 

UTILITY  OF  MATHEMATICS. 

CHAPTER  I. 

The  Utility  of  Mathematics  considered  as  a  Means  op 
Intellectual  Training  and  Culture 349 

CHAPTER  II. 

The  Utility  of  Mathematics  regarded  as  a  Means  of 
Acquiring  Knowledge — Baconian  Philosophy 864 

CHAPTER  III. 

The  Utility  of  Mathematics  considered  as  furnishing 
those  Rules  of  Art  which  make  Knowledge  Practi- 
cally Effective 381 

Alphabetical  Index 89"? 


INTRODUCTION 


OBJECTS   AND  PLAN  OF  THE  WORK. 

Utility   and    Progress    are  the  two   leading     utility 
ideas  of  the  present  age.     They  were  manifested    Progress: 
in  the  formation  of  our  political  and  social  insti-  Their  infii!- 
tutions,  and  have  been  further  developed  ni  the     emmem; 
extension  of  those  institutions,  with  their  subdu- 
ing and  civilizing  influences,  over  the  fairest  por- 
tions of  a  great  continent.     They  are  now  be- 
coming the  controlling  elements  in  our  systems  in  educatioa 
of  pubhc  instruction, 


What,  then,  must  be  the  basis  of  that  system      vvhai 

the  basis  of 

of  education  which  shall  embrace  within  its  ho-  utility  and 
rizon  a  Utility  as  comprehensive  and  a  Progress 
as  permanent  as  the  ordinations  of  Providence, 
exhibited  in  the  laws  of  nature,  as  made  known 
by  science  ?  It  must  obviously  be  laid  in  the 
examination   and   analysis   of  those  laws ;    and 


12  INTRODUCTION. 

Preparatory  primarily,  in  those  preparatory  studies  which  fit 
and  quaUfy  the  mind  for  such  Divine  Contem- 
plations. 

Bacon's  When  Bacon  had  analyzed  the  philosophy  oi 

Philosophy. 

the  ancients,  he  found  it  speculative.  The  great 
highways  of  life  had  been  deserted.  Nature, 
spread  out  to  the  intelligence  of  man,  in  all  the 
minuteness  and  generality  of  its  laws — in  all  the 
harmony  and  beauty  which  those  laws  develop — 
had  scarcely  been  consulted  by  the  ancient  phi- 
Phiioso-     losophers.      They    had    looked   within,    and    not 

phy  of  the 

Ancients,  without.  They  sought  to  rear  systems  on  the 
uncertain  foundations  of  human  hypothesis  and 
speculation,  instead  of  resting  them  on  the  im- 
mutable laws  of  Providence,  as  manifested  in 
the  material  world.  Bacon  broke  the  bars  ol 
this  mental  prison-house:  bade  the  mind  go  free, 
and  investigate  nature. 

Foundations        Bacou  laid  the  foundations  of  his  philosophy  m 
of  Bacon's    Qy„g^y;^[Q  Jaws,  and  explained  the  several  processes 

Philosophy :        o  '  r  i 

of  experience,   observation,  experiment,  and  in- 
duction, by  which   these  laws  are  made  known. 
Why  op-     He  rejected  the  reasonings  of  Aristotle  because 

pisedtoAris-     ,  •  t  r  ^       ^ 

totie's.  they  were  not  progressive  and  useiul ;  because 
they  added  little  to  knowledge,  and  contributed 
nothing  to  ameliorate  the  sufferings  and  elevate 
the  condition  of  humanity. 


PLAN     OF     THE     WORK.  13 


The  time  seems  now  to  be  at  hand  when  the     Practical 
philosophy  of  Bacon  is  to  find  its  full  develop- 
ment.    The  only  fear  is,  that  in  passing  from  a 
speculative   to  a  practical  philosophy,  we  may, 
for  a  time,  lose  sight   of  the  fact,  that  Practice 

without   Science  is    Empiricism;    and    that    all    its  true  na- 
ture. 
which  is  truly  great  in  the  practical  must  be  the 

application  and  result  of  an  antecedent  ideal. 

What,   then,  are  the   sources   of  that  Utility,     what  is 

the  true  sys 

and  the  basis  of  that  Practical,  which  the  pres-  temoiedu- 
ent  generation  desire,  and  aftei  which  they  are 
so  anxiously  seeking  ?  What  system  of  training 
and  discipline  will  best  develop  and  steady  the 
intellect  of  the  young  ;  give  vigor  and  expan- 
sion to  thought,  anid  stability  to  action?     What  which wm 

develop  and 

course  of  study  will  most  enlarge  the  sphere  of  steady  tuo 
investigation  ;  give  the  greatest  freedom  to  the 
mind  without  licentiousness,  and  the  greatest 
frieedom  to  action  consistent  with  the  laws  of 
nature,  and  the  obligations  of  the  social  com- 
pact ?     What  subject  of  study  is,  from  its  na-    what  are 

.     .  the  subjects 

ture,   most   likely  to   ensure   this    training,  and    of  study? 
contribute  to  such  results,  and  at  the  same  time 
lay  the  foundations  of  all  that   is   truly  great  in 
the  Practical  ?     It  has  seemed  to  me  that  math-  Mathematica 
ematical  science  may  lay  claim  to  this  pre-emi- 
nence. 


14 


INTRODUCTION 


Founda-  The  fii'st  jmpressions  which  the  child  receives 

tionsofmath-       r-    tv-         i  i    r\  •  j  r  i       •  l 

ematicai  ^^  iNumbcr  and  (4uantity  are  the  loundations  oi 
Lnovviedge.  j^j^  mathematical  knowledge.  They  form,  as  it 
were,  a  part  of  his  intellectual  being.  The  laws 
Laws  of  of  Nature  are  merely  truths  or  generalized  facts, 
in  regard  to  matter,  derived  by  induction  from 
experience,  observation,  and  experiment.  The 
laws  of  mathematical  science  are  generalized 
truths  derived  from  the  consideration  of  Number 
and  Space.  All  the  processes  of  inquiry  and 
investigation  are  conducted  according  to  fixed 
laws,  and  form  a  science  ;  and  every  new  thought 
and  higher  impression  form  additional  links  in 
the  lengthening  chain. 


Nature. 


Number 

and 
tspace. 


Mathemat- 
ical kiiowl- 
ndge  : 


What  it 
does. 


The  knowledge  which  mathematical  science 
imparts  to  the  mind  is  deep — profound — abiding. 
It  gives  rise  to  trains  of  thought,  which  are  born 
in  the  pure  ideal,  and  fed  and  nurtured  by  an 
acquaintance  with  physical  nature  in  all  its  mi- 
nuteness and  in  all  its  grandeur :  which  survey 
the  laws  of  elementary  organization,  by  the  mi- 
croscope, and  weigh  the  spheres  in  the  balance 
of  universal  gravitation. 


What  '^^^  processes  of  mathematical  science  serve 

the  processes  j-q  gjyg  mental  uuity  and  wholeness.     They  im- 
part that  knowledge  which  applies  the  means  of 


PLANOFTHEWORK.  15 

crystallization  to  a  chaos  of  scattered  particulars,  Right  knowi- 
and  discovers  at  once  the  general  law,  if  there  t,,g  ,^^^^3  ^^^ 
be  one,  which  forms  a  connectino;  link  between    "J's'a"'2a- 

'  o  lion, 

them.  Such  results  can  only  be  attained  by 
minds  highly  disciplined  by  scientific  combina- 
tions. In  all  these  processes  no  fact  of  science 
is  forgotten  or  lost.  They  are  all  engraved  on 
the  great  tablet  of  universal  truth,  there  to  be 
read  by  succeeding  generations  so  long  as  the  it  records 
laws  of  mind  remain  unchanged.  This  is  stri-  truth, 
kingly  illustrated  by  the  fact,  that  any  diligent 
student  of  a  college  may  now  read  the  works  of 
Newton,  or  the  Mecanique  Celeste  of  La  Place. 

The   educator   regards   mathematical   science     How  uu^ 

educator  re- 

as  the  great  means  of  accomplishing  his  work,  gardsmatii- 
The  definitions  present  clear  and  separate  ideas, 
which  the  mind  readily  apprehends.  The  axioms  xheaxioma. 
are  the  simplest  exercises  of  the  reasoning  fac- 
ulty, and  afford  the  most  satisfactory  results  in 
the  early  use  and  employment  of  that  faculty. 
The  trains  of  reasoning  which  follow  are  com- 
binations, according  to  logical  rules,  of  what 
has  bfeen  previously    fully    comprehended;    and  influence ot 

.  ,  the  study  of 

the  mind  and  the    argument  grow   together,  so  mathematics 
that  the  thread  of  science  and  the  warp  of  the  °" '  "  *"'" 
intellect  entwine  themselves,  and  become  insep- 
arable.    Such  a  training  will  lay  the  foundations 


10  INTRODUCTION. 


of  systematic  knowledge,  so  greatly  preferable 
to  conjectural  judgments. 

How  the         The  philosopher  regards  mathematical  science 

philosopher  i         r   i  •      i  •    i  •  t        i 

regards        ^^  the  mCrC  tOOlS  01    hlS    higher  vocation.        Look- 
mathematics:    •  •,1  iJ  J  •  iAT. 

ing  with  a  steady  and  anxious  eye  to  JNature, 
and  the  great  laws  which  regulate  and  govern 
all  things,  he  becomes  earnestly  intent  on  their 
examination,  and  absorbed  in  the  wonderful  har- 
monies which  he  discovers.  Urged  forward  by 
Its  necessity  these  high  impulses,  he  sometimes  neglects   that 

to  him. 

thorough  preparation,  in  mathematical  science, 
necessary  to  success ;  and  is  not  unfrequently 
obliged,  like  Antaeus,  to  touch  again  his  mother 
earth,  in  order  to  renew  his  strength. 

The  views        The  mci'c  practical  man  regards  with    favor 

'jf  the  practi- 
cal man.     oiily  the  results  of  science,  deeming  the  reason- 
ings through  which  these  results  are  arrived  at, 
quite  superfluous.     Such  should  remember  that 

Instruments  the  mind  rcquircs    instruments    as   well    as    the 

of  the  mind  i         i  i     i  •         i     • 

hands,  and  that  it  should  be  equally  trained  in 
their  combinations  and  uses.  Such  is,  indeed, 
now  the  complication  of  human  affairs,  that  to 
do  one  thing  well,  it  is  necessary  to  know  the 
properties  and  relations  of  many  things.  Every 
Everything  thing,  whether  existing  in  the  abstract  or  in  the 

has  a  law. 

material  world;  whether  an  element  of  knowl- 


PLANOFTHEWORK.  17 


edge  or  a  rule  of  art,  has  its  connections  and  its     to  know 

,      ,  the  law  is  to 

law :  to  understand  these  connections  and  that    know  the 
law,  is  to  know  the  thing.     When  the  principle      ^^"^' 
is  clearly  apprehended,  the  practice  is  easy. 


With  these  general  views,  and  under  a  firm  Mathemanw 

,  ,  .       ,  .  ,  analyzed. 

conviction  that  mathematical  science  must  be- 
come the  great  basis  of  education,  I  have  be- 
stowed much  time  and  labor  on  its  analysis,  as 
a  subject  of  knowledge.  I  have  endeavored  to 
present  its  elements  separately,  and  in  their  con-  How. 
nections ;  to  point  out  and  note  the  mental  fac- 
ulties which  it  calls  into  exercise  ;  to  show  why 
and  how  it  develops  those  faculties ;  and  in  what 
respect  it  gives  to  the  whole  mental  machinery 
greater  power  and  certainty  of  action  than  can 
be    attained   by  other  studies.     To   accomplish    whatwaa 

deemed  ne- 

these  ends,  in  the  way  that  seemed  to  me  most     cessary. 
desirable,   I  have  divided  the  work  into   three  . 
parts,  arranged  under  the  heads  of  Book  I.,  II., 
and  III 

Book  I.  treats  of  Logic,  both  as  a  science  and      Logic 
an  art ;  that  is,  it  explains  the  laws  which  gov- 
ern  the  reasoning   faculty,  in   the   complicated 
processes  of  argumentation,  and  lays  down  the  Explanation, 
rules,  deduced  from  those  laws,  for  conducting 
such  processes.      It   being  one   of  the   leading 

2 


18  INTRODUCTION. 


For  what    objects  to  show  that  mathematical  science  is  the 

best  subject  for  the  development  and  application 

of  the  principles  of  logic  ;  and,  indeed,  that  the 

science  itself  is  but  the  application  of  those  prin- 

Tiie  necessity  ciplcs   to  the    abstract    quantities  Number    and 

of  treating  it.  _ 

Space,  it  appeared  indispensable  to  give,  in  a 
manner  best  adapted  to  my  purpose,  an  out- 
line of  the  nature  of  that  reasoning  by  means 
of  which  all  inferred  knowledge  is  acquired. 

Book  II.  Bool-    ji    treats    of    Mathematical    Science, 

Here  I  have  endeavored  to  explain  the  nature  of 
Of  what  it  the  subjects  with  which  mathematical  science  is 
conversant ;  the  ideas  which  arise  in  examining 
and  contemplating  those  subjects ;  the  language 
employed  to  express  those  ideas,  and  the  laws  of 
their  connection.  This,  of  course,  led  to  a  class- 
Manner  of    ification  of  the  subjects;  to  an  analysis  of  the 

treating.  .         .  ^    • 

language  used,  and  an  examination  of  the  reason- 
ings employed  in  the  methods  of  proof. 

Book  tii.         Book  III.  explains  and  illustrates  the  Utility  of 

utility  of      T.,      ,  .  _,.  ,     ,.       . 

Mathematics.  Mathematics  :  r  irst,  as  a  means  of  mental  disci- 
pline and  training ;  Secondly,  as  a  means  of  ac- 
quiring knowledge ;  and.  Thirdly,  as  furnishing 
those  rules  of  art,  which  make  knowledge  prac- 
tically effective. 


PLANOFTHEWORK.  19 


Having  thus  given  tiie  general  outlines  of  the    classes  of 
work,  we  will  refer  to  the  classes  of  readers  for 
whose  use  it  is  designed,  and  the  particular  ad- 
vantages and  benefits  which  each  class  naay  re- 
ceive from  its  perusal  and  study. 

There  are  four  classes  of  readers,  who  may.  Four  classes 
it  is  supposed,  be  profited,  more  or  less,  by  the 
perusal  of  this  work  : 

1st.  The  general  reader  ;  First  class. 

2d.  Professional  men  and  students  ;  second. 

3d.  Students  of  mathematics  and  philosophy  ;        ThM. 

4th.  Professional  Teachers.  Fourth. 

First.  The  general  reader,  who  reads  for  im.   Advantages 

1      1      •  ^  -1  11  to  the  gen- 

provement,  and  desires  to   acquire   knowledge,   erai  reader, 
must  carefully  search  out  the  import  of  language. 
He  must  early  establish  and  carefully  cultivate 
the  habit  of  noting  the  connection  between  ideas     connec- 
and  their  signs,  and  also  the  relation  of  ideas  to    ^ordsaud" 
each   other.     Such   analysis   leads   to  attentive       ^'^^'^^ 
reading,  to  clear    apprehension,  deep  reflection, 
and  soon  to  generalization. 

Logic  considers  the  forms  in  which  truth  must      Logir. 
be  expressed,  and  lays  down  rules  for  reducing 
all  trains  of  thought  to  such  known  forms.     This 
habit  of  analyzing  arms  us  with  tests  by  which    itsvaiue: 
we  separate  argument  from  sophistry — truth  from 
falsehood.     The  application  of  these  principles, 


20  INTRODUCTION. 


fn  the  study  in  the  Construction  of  the  mathematical  science. 

mathematics,  wherc  the  relation  between  the  sign  (or  language) 
and  the  thing  signified  (or  idea  expressed),  is  un- 
mistakable, gives  precision  and  accuracy,  leads 
to  right  arrangement  and  classification,  and  thus 
prepares  the  mind  for  the  reception  of  general 
knowledge. 

Advantages       Secondly.    The  increase  of  knowledge  carries 

to  profession-        .,.,  .  ri         •  c         •  ai-'i 

aimen.      With  it  the  ncccssity  ot  classincation.     A  limited 
number  of  isolated  facts  may  be  remembered,  or 
a  few  simple  principles  applied,  without  tracing 
out  their  connections,  or  determining  the  places 
which   they   occupy  in    the    science    of  general 
knowledge.     But  when  these  facts  and  principles 
are  greatly  multiplied,  as  they  are  in  the  learned 
The  reason,  profcssions  ;  wlicu  the  labors  of  preceding  gen- 
erations are  to  be  examined,  analyzed,  compared ; 
when  new  S3'stems  are  to  be  formed,  combining 
all  that  is  valuable  in  the  past  with  the  stimu- 
lating elements  of  the  present,  there  is  occasion 
for  the  constant  exercise  of  our  highest  facul- 
Knowiedttc  ties.       Knowledge    reduced    to    order  ;    that    is, 
reduce   to   j^i^Q^yigfjo-g   SO  classified   and  arranged   as   to  be 

order  is  o  => 


science. 


easily  remembered,  readily  referred  to,  and  ad- 
vantageously applied,  will  alone  suffice  to  sift 
the  pure  metal  from  the  dust  of  ages,  and  fashion 
it  for  present  use.     Such  knowledge  is  Science. 


PLAN     OF     THE     WORK.  21 


Masses  of  facts,  like  masses  of  matter,  are  ca-   Knowledge 
pabie  01  very  mmute  subdivisions;  and  when  we  aucedtoita 
know  the  law  of  combination,  they  are  readily    ^^'^"»'^°^ 
divided  or  reunited.     To  know  the  law,  in  any 
case,  is  to  ascend  to  the  source  ;  and  without 
that  knowledge  the  mind  gropes  in  darkness. 

It  has  been  my  aim  to  present  such   a  view    objects  of 
of  Logic    and   Mathematical   Science   as  would 
clearly  indicate,  to  the  professional  student,  and 
even  to  the  general  reader,  the  outlines  of  these 
subjects.       Logic  exhibits    the  general    formula    i^ogicaiwi 

1-11  11     1  •      1  r  •  1   mathematica 

applicable  to  ail  kinds  oi  argumentation,  and 
mathematics  is  an  application  of  logic  to  the 
abstract  quantities  Number  and  Space. 

When  the  professional  student  shall  have  ex- 
amined the  subject,  even  to  the  extent  to  which  certainty  of 
it  is  here  treated,  he  will  be  impressed  with  the 
clearness,  simplicity,  certainty,  and  generality  of 
its  principles ;  and  will  find  no  difficulty  in  ma- 
king them  available  in  classifying  the  facts,  and 
examining  the  organic  laws  which  characterize 
his  particular  department  of  knowledge. 


Thirdly.  Mathematical  knowledge  differs  from  Maihemati- 

every  other  kind  of  knowledge  in  this  :  it  is,  as  ^^  ^Z^ 
it  w'ere,  a  web  of  connected  principles  spun  out 
from  a  few  abstract  ideas,  until  it  has  become 

one  of  the  great  means  of  intellectual  develop-  its  extent. 


22  INTRODUCTION. 


ment  and  of  practical  utility.     And  if  I  am  per- 
Necessity     mitted  to  cxtcnd  the  figure,  I  may  add,  that  the 

of  beginning  t        r     \  •  i  i  i 

at  the  right  WBD  of  the  spidcr,  though  perfectly  simple,  if  we 
*''"*■  see  the  end  and  understand  the  way  in  which 
it  is  put  together,  is  yet  too  complicated  to  be 
unravelled,  unless  we  begin  at  the  right  point, 
and  observe  the  law  of  its  formation.  So  with 
mathematical  science.  It  is  evolved  from  a  few 
— a  very  few — elementary  and  intuitive  princi- 
How       pies  :  the  law  of  its  evolution  is  simple  but  ex- 

mathemati-  .  i  i        •  i  •    i  i 

cai  science  ifl  acting,  and  to  bcgm  at  the  right  place  and  pro- 
constructe  .  ^gg^  jj^  ^jjg  j-jght  way,  is  all  that  is  necessary  to 

make  the  subject  easy,  interesting,  and  useful. 
What  has        I  have  endeavored  to  point  out  the  place  of 

been  at-  _        _ 

templed,  beginning,  and  to  indicate  the  way  to  the  math- 
ematical student.  I  am  aware  that  he  is  start- 
ing on  a  road  where  the  guide-boards  resemble 
each  other,  and  where,  for  the  want  of  careful 
observation,  they  are  often  mistaken  ;  I  have 
sought,  therefore,  to  furnish  him  with  the  maps 
and  guide-books  of  an  old  traveller. 
Advantages       By  explaining  with   minuteness   the   subjects 

of  examining      ,  i  •    i  i  •       i         • 

the  whole    about  which  mathematical  science  is  conversant, 
subject,     ^j^g  whole  field  to  be  gone  over  is  at  once  sur- 
veyed :  by  calling  attention  to  the  faculties  of 
Advantages  \\^q  mind  which  the  science  brings  into  exercise, 

of  consider-  "^ 

ing  the  men-  we  are  better  prepared  to  note  the  intellectual 

tal  faculties :  ^ .  i  •    t       t  •  11 

operations  which  the  processes  require ;  and  by 


PLAN     OF     THE     WORK.  23 


a  knowledge  of  the  laws  of  reasoning,  and  an  ofaknowj- 

.   ,        ,  CI  ^t^se  of  the 

acquaintance  with  the  tests  oi  truth,  we  are  en-  lawsofrea- 
abled  to  verify  all  our  results.    These  means  have      s°'»'^- 
been  furnished  in    the   following  work,  and  to 
aid  the  student  in  classification  and  arrangement, 
diagrams  have  been  prepared  exhibiting  separ-    What  has 

been  doatii 

ately  and  in  connection  all  the  principal  parts  of 
mathematical  science.  The  student,  therefore, 
who  adopts  the  system  here  indicated,  will  find 
his  way  clearly  marked  out,  and  will  recomiise,   Advantage* 

''  J  o  '      tothesto- 

from  their  general  resemblance  to  the  descrip-  dent, 
lions,  all  the  guide-posts  which  he  meets.  He 
will  be  at  no  loss  to  discover  the  connection 
between  the  parts  of  his  subject.  Beginning 
with  first  principles  and  elementary  combina- 
tions, and  guided  by  simple  laws,  he  will  go  for-      ^^'•e^ 

he  begins. 

ward  from  the  exercises  of  Mental  Arithmetic 
to  the  higher  analysis  of  Mathematical  Science 
on  an  ascent  so  gentle,  and  with  a  progress  so      omer 

of  progress 

steady,  as  scarcely  to  note  the  changes.  And 
indeed,  why  should  he  ?  For  all  mathematical 
processes  are  alike  in  their  nature,  governed  by 
the  same  laws,  exercising  the  same  faculties,  unity  of 
and  lifting  the  mind  towards  the  same  eminence. 


the  Bubjeot, 


Fourthly.  The  leading  idea,  in  the  construe-  Advantages 


to  the 


tion  of  the  work,  has  been,  to  afford  substantial  professional 
aid  to  the  professional  teacher.     The  nature  ol     '<'^*>«'- 


24  INTRODUCTION, 


His  duties:   his  duties — their   inherent  difficulties — the  per- 
Discouragc-   plexities  which  meet  him  at  every  step — the  want 
difflcuiues:    t>f  Sympathy  and  support  in  his  hours  of  discour- 
agement —  (and   they  are  many)  —  are  circum- 
stances which  awaken   a  lively  interest  in   the 
hearts  of  all  who  have  shared  the  toils,  and  been 
themselves  laborers  in  the  same  vineyard.     He 
takes  his  place  in  the  schoolhouse  by  the  road- 
side, and  there,  removed  from  the  highways  of 
Bemotenesa  life,  Spends  his  days  in  raising  the  feeble  mind 
life.        of  childhood  to  strength — in  planting  aright  the 
seeds  of  knowledge — in  purbing  the  turbulence 
of  passion  —  in    eradicating  evil    and    inspiring 
good.     The  fruits    of  his   labors   are  seen   but 
once  in  a  generation.     The  boy  must  grow  to 
fruits  of    manhood  and  the  girl  become  a  matron  before 

hia  efforts,  .  •         i  i  •      i    i  i  i 

when  seen    he  IS  ccrtam  that  his  labors  have  not  been  m 
vain. 

Yet,  to  the  teacher  is  committed  the  high  trust 
of  forming  the  intellectual,  and,  to  a  certain  ex- 
tent, the  moral  development  of  a  people.     He 

Theimpor-  holds  in  liis  hands  the  keys  of  knowledge.  If 
labors.  ^^®  ^^'^^  moral  impressions  do  not  spring  into 
life  at  his  bidding,  he  is  at  the  source  of  the 
stream,  and  gives  direction  to  the  current.  Al- 
though himself  imprisoned  in  the  schoolhouse, 
his  influence  and  his  teachings  affect  all  condi- 
tions of  society,  and  reach  over  the  whole  hori- 


PLAN     OF     THE     WORK. 


25 


zon   of  civilization.      He  impresses  himself  on  The  influence 
the  young  of   the  age  in  which  he   lives,   and 
lives  again  in  the  age  which  succeeds  him. 


All  good  teaching  must   flow  from   copious   scurcesof 

good  teach- 

knowledge.     The  shallow  fountain  cannot  emit        ing. 
a  vigorous  stream.     In  the  hope  of  doing  some- 
thing  that    may    be    useful    to    the    professional 
teacher,   I    have    attempted    a   careful    and   full    objects  for 

which  the 

analysis  of  mathematical  science.     I  have  spread    work  was 

undcrtakeo. 

out,  in  detail,  those  methods  which  have  been 
carefully  examined  and  subjected  to  the  test  of 
long  experience.      If  they  are  the  right  meth-    principles 
ods,  they  will  serve   as   standards  of  teaching ;  .°  \  ^^^ 

'J  *=>  '   ing,  the  same 

for,  the   principles  of  imparting  instruction  are 
the  same  for  all  branches  of  knowledge. 


The   system  which  I  have  indicated   is  com-      System, 
plete  in  itself     It  lays  open  to  the  teacher  the 
entire  skeleton  of   the  science — exhibits  all  its     ^'^hat  u 

presents. 

parts   separately   and  in   their   connection.      It 
explains  a    course  of  reasoning  simple  in  itself,     what  u 
and  applicable    not    only    to    every    process    in     **p'''"* 
mathematical    science,    but   to   all    processes    of 
argumentation  in  every  subject  of  knowledge. 

The  teacher  who  thus  combines  science  with      science 
art,  no   longer   regards   Arithmetic  as    a   mere    <»'nWned 

^  °  with  art: 

treadmill  of  mechanical  labor,  but  as  a  means — 


26  INTRODUCTION. 


Theadvan-   and  the  simplest  means — of  teaching  the  art  and 

taefts  result-         .  r  •  a.- t.  i     ii_*       • 

iM  from  it.  science  of  reasonmg  on  quantity — and  this  is 
the  logic  of  mathematics.     If  he  would  accom- 

Resuiuof  piish  well  his  work,  he  must  so  instruct  his 
uon.  pupils  that  they  shall  apprehend  clearly,  think 
quickly  and  correctly,  reason  justly,  and  above 
all,  he  must  inspire  them  with  a  love  of  knowl- 
edge. 


BOOK     I. 

LOGIC. 


CHAPTER    I. 

OEFINiTION'S— OPERATIONS  OF  THE  MIND ^TERMS  DEFINKD. 

DEFINITIONS. 

§  1.   Defivition  is  a  metaphorical  word,  which    Definition 

a 

literally    signifies    "  laying   down    a   boundary."  metaphorical 
All  definitions  are  of  names,  and  of  names  only ; 

Some 

but  in  some  definitions,  it  is  clearly  apparent,  definitions 
that  nothing  is  intended  except  to  explain  the      ''oniy'° 
meaning  of  the  word;  while  in  others,  besides      '^o'^'^- 
explaining  the  meaning  of  the  word,  it  is  also      ^;j,„^j 

implied  that  there  exists,  or  may  exist,  a  thinff  «>'"respond- 

^  '  "^  =•       ingtothe 

corresponding  to  the  word.  words. 


§  2.  Definitions  which  do  not  imply  the  exist-  or  definitions 

ence  of  things  corresponding  to   the  words  de-  not  imply 

fined,  are  those  usually  found  in  the  Dictionary  """^  ^°"^ 

•'                                                            J  spondmg 

of  one's  own  language.     They  explain  only  the  to  words. 


28 


LOGIC. 


[book  I. 


,j.  meaning  of  the  word  or  term,  by  giving  some 

explain     equivalent  expression  which  may  happen  to  be 

words  by  '  ^  ^  ri 

equivalents,  better  known.  Definitions  which  imply  the  ex- 
istence of  things  corresponding  to  the  words  de- 
fined, do  more  than  this. 

For  example :  "  A  triangle  is  a  rectilineal  fig- 
ure having  three  sides."  This  definition  does 
two  things : 

1st.  It  explains  the  meaning  of  the  word  tri- 
angle ;  and, 

2d.  It  implies  that  there  exists,  or  may  exist, 
a  rectilineal  figure  having  three  sides. 


Definition 

of  a 

triangle ; 

what 

it 

implies. 


ofa  §  3.  To  define  a  word  when  the  definition  is 

definition 

which  im-   to   imply  the  existence  of  a  thing,  is   to  select 
^  iTence  of'  fi'O"^  ^11  the  properties  of  the  thing  those  which 
a  thing.      „  j,g  i^Qgt  simple,  general,  and  obvious ;  and  the 
Properties    properties  must  be  very  well  known  to  us  before 
known.      ^^®  ^^^  decide  which  are  the  fittest  for  this  pur- 
pose.    Hence,  a  thing  may  have  many  properties 
besides  those  which  are  named  in  the  definition 
A  definition  of  the  word  which  stands'  for  it.     This  second 
truth.       l^ii^^  o^  definition  is  not  only  the  best  form  of  ex- 
pressing certain  conceptions,  but  also  contributes 
to  the  development  and  support  of  new  truths. 


to  §  4.  In  Mathematics,  and  indeed,  in  all  strict 

Mathematics 

names  imply  scienccs,  namcs  imply  the  existence  of  the  things 


CHAP.  I.]  DKFINITIONS.  29 

which  they  name ;  and  the  definitions  of  those      things 
names  express  attributes  of  the  things ;  so   that      express 

,'£••.•  I      ,  c  iU  attributes 

no  correct  aennition  whatever,  oi    any  mathe- 
matical  term,   can  be  devised,  which  shall   not 
express  certain  attributes  of  the  thing  correspond- 
ing to  the  name.     Every  definition  of  this  class    definitions 
is  a  tacit  assumption  of  some  proposition  which  °''""^<^i'^' 
is    expressed    by    means    of   the    definition,  and  propositions. 
which  gives  to  such  definition  its  importance. 


§  5.  All  the  reasonings  in  mathematics,  which    Keasoning 
rest   ultimately  on    definitions,   do,  in  fact,  rest  /'f"."^°" 

•'  denmtiona; 

on   the   intuitive    inference,    that   things    corre- 

rests  on 

spondmg  to  the  words  defined  have   a  conceiv-     intuitive 
able  existence  as  subjects  of  thought,  and  do  or 
may  have  proximately,  an  actual  existence.* 


*  There  are  four  rules  which  aid  us  in  framing  defini-    p-our  rules 
tions. 

1st.  The  definition  must  be  adequate:  that  is,  neither  too     1st  rule, 
extended,  nor  too  narrow  for  the  word  defined. 

2d.  The  definition  must  be  in  itself  plainer  than  the  word      id  rule, 
defined,  else  it  would  not  explain  it. 

3d.  The  definition  should  be  expressed  in  a  convenient     -^^  ^iig, 
number  of  appropriate  woi-ds, 

4th.  When  the  definition  implies  the  existence  of  a  thing 
corresponding  to  the  word  defined,  the  certainty  of  lliat 
existence  must  be  intuitive. 


30  LOGIC.  [book  I. 


OPERATIONS  OF  THE  MIND  CONCERNED  IN  REASONING. 

Three  opera-      §  Q    There  are  three  operations  of  the  mind 
dons  of  the  which  are  immediately  concerned  in  reasoning. 
1st.    Simple    apprehension  ;    2d.    Judgment ; 
3d.  Reasoning  or  Discourse. 

Sim  lea  ^  '^-   Simple   apprehension   is   the   notion    (or 

prehension,  conception)  of  an  object  in  the  mind,  analogous 
to  the  perception  of  the  senses.  It  is  either 
incompicx.  Incomplcx  or  Complex.  Incomplex  Apprehen- 
sion is  of  one  object,  or  of  several  without  any 
relation  being  perceived  between  them,  as  of  a 
Complex,  triangle,  a  square,  or  a  circle :  Coiuplex  is  ot 
several  with  such  a  relation,  as  of  Zi  triangle 
within  a  circle,  or  a  circle  within  a  siju^rc. 

§  8.  Judgment  is  the  compariur;  together  in 
the  mind  two  of  the  notions  Cor  ideas)  which 

Judgment  ^ 

defined,     are  the  objects  of  apprehension,  whether  com- 
plex or  incomplex,   and  pronouncing  that  they 
agree  or  disagree  with  each  other,  or  that  one 
of  them  belongs  or  does  not  belCj/  to,  the  other: 
for  example :  that  a  right-arg'od  t-^iangle  and  an 
Judgment    equilateral  triangle  belong  to  thf>  r'^ass  of  figures 
either      Called  triangles  ;  or  that  a  iou'.rP  is  not  a  circle, 
rma  i\e    j^(jgfyj,gjj^^  therefore,  is  eitiipr  Affirmative  or  Neg 

negative.      ^^j-j,g 


CHAP.  1  J  ABSTRACnON,  31 


§  9.    Reasoning   (or  discourse)   is   the   act  of  neasoning 
proceeding  from  certain  judgments   to  another 
founded  upon  them  (or  the  result  of  them). 


§10.    Language  affords  the  signs  by  which    Language 


aSb  rds 
signs  of 


these  operations  of  the  mind  are  recorded,  ex- 
pressed, and  communicated.     It  is  also  an  in-    thought: 
strument  of  thought,   and  one  of  the  principal     also,  an 

instrument 

helps  in  all  mental  operations ;  and  any  imper-    of  thought, 
fection   in    the   instrument,  or  in  the   mode  of 
using  it,  will  materially  affect  any  result  attained 
through  its  aid. 


§  11.  Every  branch  of  knowledge  has,   to  a 

Every  branch 

certain   extent,  its   own   appropriate   language ;  ofknowiedge 

1     r  -1  -1  1     •  I  has  its  own 

and  lor  a   mmd    not  previously  versed  m  the    language, 

meaning  and  right  use  of  the  various  words  and 

signs  which  constitute  the  language,  to  attempt     must  be 

learned. 

the  study  of  metiiods  of  philosophizing,  would 


which 


be  as  absurd  as  to  attempt  reading  before  learn- 
ing the  alphabet. 


ABSTRACTION. 


§  12.  The  faculty  of  abstraction  is  that  power 
of  the  mind  which  enables  us,  in  contemplating 
any  object  (or  objects),  to  attend  exclusively  to 


Abstraction, 


32  LOGIC.  [book  I, 

some  particular  circumstance  belonging  to  it,  and 
quite  withhold  our  attention  from  the  rest.    Thus, 

in  . 

contempia-   if  a  pcrson  in  contemplating  a  rose  should  make 

"      "  '  the  scent  a  distinct  object  of  attention,  and  lay 

aside    all   thought   of  the  form,    color,    &c.,   he 

would  draw  off",  or  abstract  that  particular  part ; 

of  drawing  and  therefore  employ  the  faculty  of  abstraction. 
He  would  also  employ  the  same  faculty  in  con- 
sidering whiteness,  softness,  virtue,  existence,  as 
entirely  separate  from  particular  objects. 

§  13.    The   term  abstraction,  is   also  used   to 
denote  the  operation  of  abstracting  from  one  or 

The  term  -'  ° 

tbstraciion,  more  things  the  particular  part  under  consider- 

hc'v  used.  .  . 

ation ;  and  likewise  to  designate  the  state  oi  the 
mind  when  occupied  by  abstract  ideas.     Hence, 
abstraction  is  used  in  three  senses : 
Abstraction       ^^^-    '^^   denote    a   faculty  or   power  of  the 
^?"T     mind ; 

a  faculty,  ' 

a  process,        gd.  To  dcnotc  a  process  of  the  mind ;  and, 

and  a  state  * 

of  mind.         3(j^  To  denote  a  state  of  the  mind. 


GENERALIZATION. 


General  Iza- 


§  14.    Generalization   is   the  process   of  con- 
tion— the    templating  the  agreement  of  several  objects  in 

process  of 

contempia-  certain    points  (that  is,  abstracting  the  circum- 

ting  the  _  _ 

-j«r.'(!ment.   stauccs   of   agreement,   disregarding   the   diner- 


CHAP.  I.]  TERMS.  33 


of  several 
things. 


ences),  and  giving  to  all  and  each  of  these  ob- 
jects a  name  applicable  to  them  in  respect  to 
this  agreement.  For  example  ;  we  give  the 
name  of  triangle,  to  every  rectilineal  figure  hav- 
ing three  sides  :    thus  we  abstract  this  property 

P  ,  ...  Generaliza- 

from  all  the  others   (for,  the  triangle  has   three       uon 
angles,  may  be  equilateral,  or  scalene,  or  right- 
angled),  and  name  the  entire  class  from  the  prop- 
erty   so    abstracted.      Generalization    therefore     imphM 

.,       .        ,.  ■.  .  ,  ,        ,  abstraction. 

necessarily  implies  abstraction ;  tnough  abstrac- 
tion does  not  imply  generalization. 

TERMS SINGULAR  TERMS COMMON  TERMS. 

§  15.    An  act  of  apprehension,   expressed   in 
language,  is  called  a  Term.     Proper  names,  or     a  term, 
any  other  terms  which  denote  each  but  a  single 
individual,    as    "  Caesar,"    "  the    Hudson,"    "  the 
Conqueror   of    Pompey,"    are    called    Singular     singular 

terms. 

Terms. 

On  the  other  hand,  those  terms  which  denote 
any  individual  of  a  whole  class  (which  are  form- 
ed by  the  process  of  abstraction  and  generaliza- 
tion), are  called  Common  or  general  Terms.    For     commoD 

termd. 

example ;  quadrilateral  is  a  common  term,  appli- 
cable to  every  rectilineal  plane  figure  having 
four  sides  ;  River,  to  all  rivers ;  and  Conqueror, 
to  all  conquerors.  The  individuals  for  which  a 
common  term  stands,  are  called  its  Signijicates.     signiflcates 

3 


34 


LOGIC. 


[book  1. 


CLASSIFICATION. 


Genus, 
species. 


Examples 


classification. 


^    -a  .•  §  16.  Common  terms  afford  the  means  of  clas- 

Classiflcation.         ' 

sification ;  that  is,  of  the  arrangement  of  objects 
into  classes,  with  reference  to  some  common  and 
distinguishing  characteristic.  A  collection,  com- 
prehending a  number  of  objects,  so  arranged,  is 
called  a  Genus  or  Species — genus  being  the 
more  extensive  term,  and  often  embracing  many 
species. 

For  example :  animal  is  a  genus  embracing 
every  thing  vv^hich  is  endowed  with  life,  the  pow- 
er of  voluntary  motion,  and  sensation.  It  has 
many  species,  such  as  man,  beast,  bird,  &c.  II 
we  say  of  an  animal,  that  it  is  rational,  it  be 
longs  to  the  species  man,  for  this  is  the  charac- 
teristic of  that  species.  If  we  say  that  it  has 
wings,  it  belongs  to  the  species  bird,  for  this,  in 
like  manner,  is  the  characteristic  of  the  species 
bird. 

A  species  may  likewise  be  divided  into  classes, 
or  subspecies ;  thus  the  species  man,  may  be 
divided  into  the  classes,  male  and  female,  and 
these  classes  may  be  again  divided  until  we  reach 
the  individuals. 


Subspecies 


Principles        §  17.  Now,  it  will  appear  from  the  principles 

of 

jiassification.  which  govcm  this  system  of  classification,  that 


'.   1.]  CLASSIFICATION.  35 


the  characteristic  of  a  genus  is  of  a  more  exten-   cenusmore 

.^         .  ,  .  ,  ^  .  extensive 

sive    signihcation,    but    involves    lewer   particu-  th;m  spodea, 
lars  than  that  ot  a  species.     In  Hke  manner,  the 
characteristic  of  a  species  is  more  extensive,  but 
less  full  and  complete,  than  that  of  a  subspecies  ^"'  i^ss  fuii 
or  class,  and  the  characteristics  of  these  less  full    complete. 
than  that  of  an  individual. 

For  example  ;  if  we  take  as  a  genus  the  Quadri- 
laterals of  Geometry,  of  which  the  characteristic 
is,  that  they  have  four  sides,  then  every  plane 
rectilineal  figure,  having  four  sides,  will  fail  under 
this  class.  If,  then,  we  divide  all  quadrilaterals 
into  two  species,  viz.  those  whose  opposite  sides, 
taken  two  and  two,  are  not  parallel,  and  those 
whose  opposite  sides,  taken  two  and  two,  are 
parallel,  we  shall  have  in  the  first  class,  all  irreg- 
ular quadrilaterals,  including  the  trapezoid  (1  and 
2) ;  and  in  the  other,  the  parallelogram,  the  rhom- 
bus, the  rectangle,  and  the  square  (3,4,  5,  and  6). 

If,  then,  we  divide  the  first  species  into  two 
subspecies  or  classes,  we  shall  have  in  the  one,  the 
irregular  quadrilaterals  (1),  and  in  the  other,  the 
trapezoids  (2) ;  and  each  of  these  classes,  being 
made  up  of  individuals  having  the  same  char- 
acteristics, are  not  susceptible  of  further  division. 
If  we  divide  the  second  species  into  two 
classes,  arranging  those  which  have  oblique  an- 
gles in   the   one,  and  those   which    have   right 


36 


LOGIC. 


[book  1. 


Species 
and 

classes. 


angles  in  the  other,  we  shall  have  in  the  first, 
two  varieties,  viz.  the  common  parallelogram 
and  the  equilateral  parallelogram  or  rhombus  (3 
and  4)  ;  and  in  the  second,  two  varieties  also, 
viz.  the  rectangle  and  the  square  (5  and  6). 

Now,  each  of  these  six  figures  is  a  quadri- 
lateral;  and  hence,  possesses  the  characteristic 
of  the  genus ;  and  each  variety  of  both  species 


Each  indi- 
vidual falling 

under  tlie 
genus  enjoys 

all  the      enjoys  all  the  characteristics   of  the  species  to 

characteris- 
tics. 


which  it  belongs,  together  with  some  other  dis- 
tinguishing feature ;  and  similarly,  of  all  classi- 
fications. 


Subaltern 
genus. 


Parallelo- 
gram. 


§  18.  In  special  classifications,  it  is  often  not 
necessary  to  begin  with  the  most  general  char- 
acteristics; and  then  the  genus  with  which  we 
begin,  is  in  fact  but  a  species  of  a  more  extended 
classification,  and  is  called  a  Subaltern  Genus. 

For  example ;  if  we  begin  with  the  genus  Par- 
allelogram, we  shall  at  once  have  two  species, 
viz.  those  parallelograms  whose  angles  are  oblique 
and  those  whose  angles  are  right  angles ;  and  in 
each  species  there  will  be  two  varieties,  viz.  in  the 
first,  the  common  parallelogram  and  the  rhom- 
bus ;  and  in  the  second,  the  rectangle  and  square. 


§  19.    A   genus  which  cannot   be  considered 

Highest  ° 

genus.      as  a  species,  that  is,  which  cannot  be  referred 


CHAP.   I.]  NATURE     OF     COMMON     TERMS.  37 

to  a  more  extended  classification,  is  called  the     Highest 
highest  genus ;  and  a  species  which  cannot  be 

Lowest 

considered  as  a  genus,  because  it  contains  only     species, 
individuals    having    the    same    characteristic,    is 
called  the  lowest  species. 

NATURE  OF  COMJIGN  TERMS. 


§  20.  It  should  be  steadily  kept  in  mind,  that 
the  "  common  terms"  employed  in  classification, 
have  not,  as  the  names  of  individuals  have,  any 


A  common 

term  has 
no  real  thing 

real  existing  thing  in  nature    corresponding  to  co''i''«po'Jd- 

them ;  but  that  each  is  merely  a  name  denoting 

a   certain    inadequate    notion  which   our  minds    inadequate 

have  formed  of  an  individual.     But  as  this  name 

does  not  include   any  thing  wherein   that   indi-     does  not 

.  ^-,  .       include  any 

vidual  diners  from   others  of  the   same  class,  it     thing  in 
is  applicable  equally  well  to  all  or  any  of  them,    individuals 
Thus,  quadrilateral   denotes   no   real  thing,  dis-       '^'^'"'» 
tinct  from  each  individual,  but  merely  any  recti- 
lineal figure  of  four  sides,  viewed  inadequately ; 
that  is,   after  abstracting  and  omitting  all   that 
is  peculiar  to  each  individual  of  the  class.     By 

^  ■^  but  is 

this  means,  a  common  term  becomes  applicable  applicable  to 
alike  to  any  one  of  several  individuals,  or,  taken  individuals. 
in  the  plural,  to  several  individuals  together. 

Much  needless  difficulty  has   been  raised  re-     „   ^, 

-'  Needless 

spec  ting  the  results  of  this  process :  many  hav-    difficulty. 
ing  contended,  and  perhaps  more  having  taken 


38  LOGIC.  [book  I, 

Difficulty  in  it  for  granted,  that   there   must  be  some  really 

the  interpre-         .      .  ,  .  ,.  i  /■    ^i 

tationof     existmg  thing  corresponding  to   each    ol    those 
'T™-™""     common  terms,  and  of  which  such  term  is  the 
name,  standing  for  and  representing  it.     For  ex- 
ample ;  since  there  is  a  really  existing  thing  cor- 
Noone      responding  to   and   signified  by  the   proper  and 
real  thing    gj^jg^jg^j.  ^amc   "  iEtua,"   it    has    been    supposed 

correspond-  o  ^  ^ 

ins  to  each.  |]-^^|.  the  common  term  "Mountain"  must  have 
some  one  really  existing  thing  corresponding  to 
it,  and  of  course  distinct  from  each  individual 
mountain,  yet  existing  in  each,  since  the  term, 
being  common,  is  applicable,  separately,  to  every 
one  of  them. 

The  fact  is,  the  notion  expressed  by  a  common 

term  is  merely   an  inadequate    (or  incomplete) 

Merely  an    j^q^jqj^  q|-  ^^j  individual ;  and  from  the  very  cir- 

inadequate  •' 

notion  pai-    cumstaucc  of  its  inadequacy,  it  will  apply  equally 

tiallyde-  i         ■j  .  i    ^        x         ^ 

signaling     ^ygU  ^q  any  ouc  of  scvcral  individuals.     For  ex- 

the  thing. 

ample ;  if  I  omit  the  mention  and  the  consider- 
ation of  every  circumstance  which  distinguishes 
iEtna  from  any  other  mountain,  I  then  form  a 
notion,  that  inadequately  designates  iEtna.  This 
"Mountain"  Hotion  is  expressed  by  the  common  term  "  moun- 

**         tain,"  which  does  not  imply  any  of  the  peculiar- 
applicable  '  I.  J       J  I 

^oaii       ities  of  the  mountain  ^Etna,  and  is  equally  ap- 

tnountains. 

plicable  to  any  one  of  several  individuals. 

In  regard  to  classification,  we  should  also  bear 
in  mind,  that  we    may   fix,    arbitrarily,    on    the 


CHAP,  ij  SCIENCE.  39 

characteristic  which  we  choose  to  abstract  and    May  axon 

attributes 

consider  as  the  basis  of  our  classification,  disre-    arbitraruy 
garding  all  the  rest :  so  that  the  same  individual  giassjflcation 
may  be  referred  to  any  of  several  different   spe- 
cies, and  the  same  species  to  several  genera,  as 
suits  our  purpose. 

SCIENCE. 

§  21.    Science,    in    its    popular    signification, 
means  knowledge.*     In  a  more  restricted  sense,      science 

in  its  general 

it  means  knowledge  reduced  to  order ;  that  is,       sense. 
knowledge   so  classified  and   arranged  as   to  be 
easily  remembered,  readily  referred  to,  and  ad-       Has  a 
vantageously    applied.       In    a   more    strict    and  gigi^fl"ation. 
technical  sense,  it  has  another  signification. 

"  Every  thing  in  nature,  as  well  in  the  in- 
animate as  in  the  animated  world,  happens  or 
is  done  according  to  rules,  though  we  do  not 
always  know  them.  Water  falls  according  to 
the  laws  of  gravitation,  and  the  motion  of  walk-  ®°'^'^  "^^^ 
ing  is  performed  by  animals  according  to  rules. 
The  fish  in  the  water,  the  bird  in  the  air,  move 
according  to  rules.  There  is  nowhere  any  want 
of  rule.     When  we  think  we  find  that  want,  we 

Nowhere 

can  only  say  that,  in  this  case,  the  rules  are  un-   any  warn  of 

rule. 

known  to  us.  f 

Assuming  that  all  the  phenomena  of  nature 


Views  of 
K>mt. 


*  Section  23.  f  Kant. 


40  LOGIC.  [bock  I 

Science      are  consequences  of  general  and  immutable  laws, 

a  technical    we  may  define   Science    to   be   the    analysis    of 

sense  e  ne  .  ^j^^g^  laws, — Comprehending  not  only   the    con- 

an analysis    nected  Drocesses    of -experiment    and  reasoning 

of  the  laws  r  r  o 

of  nature,  which  make  them  known  to  man,  but  also  those 
processes  of  reasoning  which  make  known  their 
individual  and  concurrent  operation  in  the  de- 
velopment of  individual  phenomena. 


§  22.  Art  is  the  application  of  knowledge  to 
.vrt,        practice.     Science  is   conversant   about  knowl- 

upphcation   q^^^q  ■  ^j.^  jg  ^he  usc  or  application  of  knowl- 
of  ^  ^  ^ 

science,      edge,  and  is  conversant  about  works.     Science 

has  knowledge  for  its  object :  Art  has  knowledge 

for  its  guide.     A  principle  of  science,  when  ap- 

phed,  becomes  a  rule  of  art.     The  developments 

of  science  increase  knowledge :  the  applications 

of  art  add  to  works.     Art,  necessarily,  presup- 

presupposes  pQses  knowledge  :  art,  in  any  but  its  infant  state, 

knowledge.  o  ^ 

presupposes  scientific  knowledge ;  and  if  every 
art  does  not  bear  the  name  of  the  science  on 
which  it  rests,  it  is  only  because  several  sciences 
are  often  necessary  to  form  the  groundwork  oi 
a  single  art.  Such  is  the  complication  of  hu- 
mustbe^*  man  afiairs,  that  to  enable  one  thing  to  be  done, 


known  be- 
ore  one  can 

be  done,     erties  of  many  things. 


it  is  often  requisite  to  know  the  nature  and  prop 

fore  one  can 


CHAP.  II.]  KNOWLEDGE.  41 


CHAPTER    II. 


SOURCES  AND  MEANS  OF  KNOWLEDGE — INDUCTION 


KNOWLEGDE. 

§  23.  Knowledge  is  a  clear  and  certain  con-   Knowledge 

j^     ,  ....  ....  ,  a  clear  con- 

ception 01  that  which  is  true,  and  implies  three    ceptionof 

things :  '^*"''  ^' "'"'' 

1st.  Firm  belief;    2d.  Of  what  is  true;    and,    impiies- 

3d.  On  sufficient  grounds.  belief- 

If  any  one,  for  example,  is  in  doiiht  respecting  -''•  ^fwhat 

one  of   Legendre's    Demonstrations,    he    cannot     3d.  on 

be  said  to  know  the  proposition  proved  by  it.     If,     ^"^  *^"^" 

r      r  r  j  grounds. 

again,  he  is  fully  co7ivinced  of  any  thing  that  is 
not  true,  he  is  mistaken  in  supposing  himself  to 
know  it ;  and  lastly,  if  two  persons  are  each  fullj/ 
conjidenl,  one  that  the  moon  is  inhabited,  and 
the  other  that  it  is  not  (though  one  of  these 
opinions  must  be  true),  neither  of  them  could 
properly  be  said  to  know  the  truth,  since  he 
cannot  have  sufficient  proof  of  it. 


Examples. 


4'.4  LOGIC.  [book  I 


FACTS     AND     TRUTHS. 

„     ...        §  24.  Our  knowledare  is  of  two  kinds  :  of  facts 

Knowledge  is         ^  O 

of  facts  and   j^^j-^j  truths.     A  fact  is  anv  thina;  that  has  been 

truths.  •'  ° 

or  IS.  That  the  sun  rose  yesterday,  is  a  fact : 
that  he  gives  Kght  to-day,  is  a  fact.  That  wa- 
ter is  fluid  and  stone  soUd,  are  facts.  We  de- 
rive our  knowledge  of  facts  through  the  medium 
of  the  senses. 
Truth  an         Truth  is  an  exact  accordance  with  what  has 

accordance 

with  what    BEEN,  IS,  Or  SHALL  BE.     There  are  two  methods 

has  been,  is,      ^  ,     .     .  ,         , 

or  shall  be.    o^  asccrtaming  truth  : 

Two  methods 
of  ascertain- 


Ist.  By  comparing  known  facts  with  each 
ins  it.       other;  and, 

2dly.  By  comparing  known  truths  with  each 
other. 

Hence,  truths  are  inferences  either  from  facts 
or  other  truths,  made  by  a  mental  process  called 
Reasoning. 

§  25.  Seeing,  then,  that  facts  and  truths  are  the 
Facts  and    elements  of  all  our  knowlcd2;e,  and  that  knowl- 

truths,  the  ^ 

elements     edge  itsclf  is  but  their  clear  apprehension,  their 

knowledge,   fii'm  belief,  and  a   distinct    conception    of  their 

relations  to  each  other,  our  main  inquiry  is.  How 

are  we  to    attain   unto    these  facts   and  truths, 

which  are  the  foundations  of  knowledge  ? 

1st.  Our  knowledge  of  facts  IS  derived  through 


CHAP.   II.]  FACTS     AND     TRUTHS.  43 


the  medium  of  our  senses,  by  observation,  exper- 
iment,* and  experience.     We  see  the  tree,  and     How  we 

s.rrive  at  a 

perceive  that  it  is  shaken  by  the  wind,  and  note  knowledge  oi 
the  fact  that  it  is  in  motion.  We  decompose 
w^ater  and  find  its  elements ;  and  hence,  learn 
from  experiment  the  fact,  that  it  is  not  a  simple 
substance.  We  experience  the  vicissitudes  of 
heat  and  cold ;  and  thus  learn  from  experience 
that  the  temperature  is  not  uniform. 

The  ascertainment  of  facts,  in  any  of  the  vi^ays 
above  indicated,  does  not  point  out  any  connec-  This  does  not 
tion  between  them.     It  merely  exhibits  them  to   connection 
the  mind  as  separate  or  isolated :  that  is,  each     ^^^^^'^^ 

^  '  them, 

as    standing   for    a   determinate    thing,    whether 
simple    or  compound.     The   term   facts,  in    the 
sense  in  which  we  shall  use   it,  will   designate 
facts  of  this  class  only.     If  the  facts  so  ascer- 
tained have  such  connections  with  each  other,   when  they 
that  additional  facts  can  be  inferred  from  them,   nectionthat 
that  inference  is  pointed  out  by  the  reasoning  'b''°h|fre°"' 
process,  which  is  carried  on,  in  all  cases,  by  com-      ®°"'"" 

process. 

parison. 

2Jly.  A  result  obtained  by  comparing  facts,  we  Truth,  ro-md 
have  designated  by  the  term  Truth.  Truths,  ^7001^^^""^ 
therefore,  are  inferences  from  facts  ;  and  every 


*  Under  this  term  we  include  all  the  methods  of  inves- 
tigation and  processes  of  arriving  at  fac  ts,  except  the  pro- 
cess of  reasoning. 


44 


LOGIC. 


[book  1 


and 
is  inferred 
from  tliern. 


How 


truth  has  reference  to  ail  the  singular  facts  from 
which  it  is  inferred.  Truths,  therefore,  are  re- 
sults deduced  from  facts,  or  from  classes  of  facts. 
Such  results,  when  obtained,  appertain  to  all  facts 
of  the  same  class.  Facts  make  a  genus  :  truths, 
a  species  ;  with  the  characteristic,  that  they  be- 
come known  to  us  by  inference  or  reasoning. 

§  26. 


truths  are      j-  /•      x       i_        ^l  •  n 

inferred  from  ifom  lacts   by  the  rcasoumg  process  r 


reasonin; 
process, 


Ist  case. 


How,  then,  are   truths  to  be  inferred 

There 

facUby  the    ^^g  ^^^  ^^ggg_ 

1st.  When  the  instances  are  so  few  and  simple 
that  the  mind  can  contemplate  all  the  facts  on 
which  the  induction  rests,  and  to  which  it  refers, 
and  can  make  the  induction  without  the  aid  of 
other  facts  ;  and, 

2dly.  When  the  facts,  being  numerous,  com- 
plicated, and  remote,  are  brought  to  mind  only 
by  processes  of  investigation. 


Qd  case. 


INTUITIVE     TRUTH. 


Intuitive 


Self-evident 
truths. 


Intuition 
defined. 


§  27.  Truths  which  become  known  by  con- 
sidering all  the  facts  on  which  they  depend,  and 
which  are  inferred  the  moment  the  facts  are 
apprehended,  are  the  subjects  of  Intuition,  and 
are  called  Intuitive  or  Self-evident  Truths.  The 
term  Intuition  is  strictly  applicable  only  to  that 
mode   of  contemplation   in  which    we   look   at 


CHAP.   II.]  INTUITIVE     TRUTH.  45 


facts,  or  classes  of  facts,  an  J  apprehend  the 
relations  of  those  facts  at  the  same  time,  and 
by  the  same  act  by  which  we  apprehend  tlie 
facts    themselves.     Hence,  intuitive  or  self-evi-  How  intuitive 

1  I  I  1-1  •         I     •  Irulhsare 

dent  truths  are  those  which   are  conceived  in  conceived  in 
the  mind  immediately ;    that  is,  which  are  per-      ^'^  ™"  " 
fectly  conceived  by   a  single  process   of  induc- 
tion, the  moment  the  facts  on  which  they  depend 
are    apprehended,    without    the    inteivention    of 
other  ideas.    They  are  necessary  consequences  of 
conceptions  respecting  which  they  are  asserted.    Axioms  of 
The  axioms  of  Geometry  afford  the  simplest  and  i^l^iJL^l^ 
most  unmistakable  class  of  such  truths.  '''"'^' 

"A  whole  is  equal  to  the  sum  of  all  its  parts,"     a  whole 

,  f.        .  ,  t      •     r  T    I-  equal  to  the 

IS  an  intuitive  or  seli-evident  truth,  interred  irom    g„mofaii 
facts  previously  learned.     For  example  ;  having    ^^'^.  ^^.^ 

I  •/  -i       '  o     an  intuitive 

learned  from  experience  and  through  the  senses  '^'^*- 
what  a  whole  is,  and,  from  experiment,  the  fact 
that  it  may  be  divided  into  parts,  the  mind  per- 
ceives the  relation  between  the  whole  and  the 
sum  of  the  parts,  viz.  that  they  are  equal ;  and 
then,  by  the  reasoning  process,  infers  that  the  now  inferred, 
same  will  be  true  of  every  other  thing;  and 
hence,  pronounces  the  general  truth,  that  "a 
whole  is  equal  to  the  sum  of  all  its  parts."  Here 
all  the  facts  from  which  the  induction  is  drawn,  -*"  ^e  facts 

are  presented 

are  presented  to    the   mind,  and  the   induction  to  the  mind, 
is  made  without  the  aid  of  other  facts ;  hence, 


4G  LOGIC.  ,  [book  1. 

Au  the      it  is  an  intuitive  or  self-evident  truth.     All  the 
deduced  in    Other    axioms   of  Geometry    are    deduced    Irom 

the  same  .  ,    ,  ^   .     ^  .      , 

^ay_  premises  and  by  processes  oi  mierence,  entn'ely 
similar.  We  would  not  call  these  experimental 
truths,  for  they  are  not  alone  the  results  of  ex- 
periment or  experience.  Experience  and  exper- 
iment furnish  the  requisite  information,  but  the 
reasoning  power  evolves  the  general  truth. 

"  When  we  say,  the  equals  of  equals  are  equal, 
we  mentally  make  comparisons  in  equal  spaces, 

These      equal   times,   &c. ;    so  that  these   axioms,   how- 
axioms  are  ,^       .  ,  ■^^  i 
general     cvcr  sclf-cvidcnt,  are   still  general  propositions : 

propositions.  ^^  ^^^  ^^  ^^^  inductivc  kind,  that,  independently 

of  experience,  they  would  not  present  themselves 

to  the  mind.     The  only  difference  between  these 

and  axioms  obtained  from  extensive  induction  is 

Diflference    this :   that,   in  raising  the   axioms  of  Geometry, 

them^and    ^^^  instauccs  offcr  thcmsclves  spontaneously,  and 

other       without  the  trouble  of  search,  and  are  few  and 

propositions, 

which  re-    simple :  in  raisins;  those  of  nature,  they  are  in- 
quire diugent         1  o  ^ 

research,  finitely  numerous,  complicated,  and  remote ;  so 
that  the  most  diligent  research  and  the  utmost 
acuteness  are  required  to  unravel  their  web,  and 
place  their  meaning  in  evidence."* 


*  Sir  John  Herschel's  Discourse  on  the  study  of  Natural 
Philosophy. 


CHAP.   II.]  LOGICAL     TRUTHS.  47 

TRUTHS,     OR     LOGICAL     TRUTHS. 

^  28.  Truths  inferred  from  facts,  by  the  process 
of  generalization,  when  the  instances  do  not  offer      Truths 

11  1  1  ■      1    1      X  ■  generalized 

themselves  spontaneously  to  the  mind,  but  require  from  facta, 
search  and  acuteness  to  discover  and  point  out  t^iu^sin- 
their  connections,   and   all   truths  inferred  from  ferredtrom 

truths. 

truths,  might  be  called  Logical  Truths.  But  as 
we  have  given  the  name  of  intuitive  or  self- 
evident  truths  to  all  inferences  in  which  all  the 
facts  were  contemplated,  we  shall  designate  all 
others  by  the  simple  term,  Truths. 

It  might  appear  of  httle  consequence  to  dis-  Necessity  or 
tinguish  the  processes  of  reasoning  by  which  ii^edistinc- 
truths  are  inferred  from  facts,  from  those  in  which  the  basis  of  :i 

classification. 

we  deduce  truths  from  other  truths ;  but  this  dif- 
ference in  the  premises,  though  seemingly  slight, 
is  nevertheless  very  important,  and  divides  the 
subject  of  logic,  as  we  shall  presently  see,  into 
two  distinct  and  very  different  branches. 


§  29.   Logic  takes  note  of  and  decides  upon       Logic 
the  sufficiency  of  the  evidence  by  which  truths  sufficiency  of 
are  established.     Our  assent  to  the   conclusion    evidence. 
being  grounded  on  the  truth  of  the  premises,  we 
never  could   arrive   at   any  knowledge  by  rea- 
soning,  unless    something  were  known    antece- 
dently  to   all  reasoning.     It  is  the   province  of  Hs  provinctt, 


48  LOGIC.  [book  I. 


Furnishes    Logic   to  fumish   the   tests  by  which  all  truths 

the  testa  of       ,  ...  i        •     r  i     r  i 

truth.       that  are  not  intuitive  may  be  inierred  irom  the 
premises.     It  has  nothing  to  do  with  ascertain- 
ing facts,  nor  with  any  proposition  which  claims 
to   be    believed    on   its  own  intrinsic  evidence ; 
that  is,  without  evidence,  in  the  proper  sense  of 
Has  nothing  the  word.     It  has  nothing  to  do  with  the  original 
intuitive  pro- '^^t^'    °^'  ultimate  premises    of  our    knowledge; 
positions,  nor  ^yj^]^  their  uumber  or  nature,  the  mode  in  which 

with  original 

data;  they  are  obtained,  or  the  tests  by  which  they 
are  distinguished.  But,  so  far  as  our  knowledge 
is  founded  on  truths  made   such   by  evidence, 

but  supplies 

all  tests  for  that  is,  derived  from  facts  or  other  truths  pre- 
propositions.  viously  kuown,  whcthcr  those  truths  be  particu- 
lar truths,  or  general  propositions,  it  is  the  prov- 
ince of  Logic  to  supply  the  tests  for  ascertaining 
the  validity  of  such  evidence,  and  whether  or 
not  a  belief  founded  on  it  would  be  well  ground- 
ed.    And   since  by  far   the  greatest  portion  of 

The  greatest  our  knowledge,  whether  of  particular  or  general 

portion  of  our  .  „   .     ^ 

knowledge    truths,   IS  avowcdly  matter  of  inference,  nearly 
comes  from  ^j^^   wholc,   not  Only  of  scicncc,  but  of  human 

inference.  •' 

conduct,  is  amenable  to  the  authority  of  logic. 


CHAP.  II.]  INDUCTION.  49 


INDTTCTION. 

§  30.  That  part  of  logic  which  infers  truths 
from  facts,  is  called  Induction.  Inductive  rea- 
soning is   the  application  of  the  reasoning  pro- 


induction, 


reasoning 

cess  to  a  given  number  of  facts,  for  the  purpose   applicable. 
of  determining  if  what  has  been  ascertained  re- 
specting one  or  more  of  the  individuals  is  true 
of   the  whole  class.      Hence,   Induction  is  not    induction 

f      1         /■  1  I       •  defined. 

the   mere   sum   of  the   facts,    but  a  conclusion 
Irawn  from  them. 
The   logic  of   Induction  consists  in   classina;    Logic  of 

.  .  .  luductioD. 

the  facts  and  stating  the  inference  m  such  a 
manner,  that  the  evidence  of  the  inference  shall 
be  most  manifest. 


§  31.  Induction,  as  above  defined,  is  a  process    induction 
of  inference.     It  proceeds  from  the  known   to 


from  the 

known  to  the 

unknown. 


the  unknown ;  and   any  operation  involving  no 

inference,  any  process  in  which  the  conclusion 

is   a  mere  fact,  and  not  a   truth,   does   not   fall 

within  the  meaning  of  the  term.     The  conclu-  The  conclu- 
sion broader 
sion  must  be  broader  than  the  premises.     The     than  the 

.  premises. 

premises  are  facts :  the   conclusion  must   be    a 
truth. 

Induction,  therefore,  is  a  process  of  general-    induction, 

T       •  1  •  r      1  -11^  process  of 

ization.     It   IS    that   operation    oi    the    mind  by   generaiizar 
which  we  infer  that  what  we  know  to  be  true 

4 


50  LOGIC.  [book  I. 


in  which     in  a  particular  case  or  cases,  will  be  true  in  all 

wo  conclude,  i  •    i  i  i       j  i        r  •  ,     • 

that  what  is  cascs  which  resemble  the  lormer  in  certani  as- 
true  under    gjnrnable  respects.     In  other  words.  Induction  is 

particular  °  '■ 

circumstan-    the   process  by  which  we   conclude    that  what 

cea  will  be      _ 

tmeuniver-  IS  truc  of  Certain  individuals  of  a  class  is  true 
of  the  whole  class ;  or  that  what  is  true  at  cer- 
tain times,  will  be  true,  under  similar  circum- 
stances, at  all  times. 

Induction        §  ^2.  luduction  always  presupposes,  not  only 
presupposes  ^^^^^  ^^^  ncccssaiT  observatious  are  made  with 

accurate  and  "^ 

necessary     the  iieccssary  accuracy,  but  also  that  the  results 

observations.       r-     i  i  •  r  •       i  i 

of  these  observations  are,  so  far  as  practicable, 
connected  together  by  general  descriptions :  ena- 
bling the  mind  to  represent  to  itself  as  wholes, 
whatever  phenomena  are  capable  of  being  so 
represented. 

To  suppose,  however,   that  nothing  more  is 

More  is     required  from  the  conception  than  that  it  should 

necessary     ggj-yg  to  conucct  the  observations,  would  be  to 

than  to 

connect  the  substitute    hypothcsis   for  theory,   and   imagina- 

observations 

we  must     tion  for  proof.     The  connecting  link  must   be 

mfer  from  i  •    i  71  •  •  ^         r 

uiem.  some  character  which  reaiLy  exists  m  the  lacts 
themselves,  and  which  would  manifest  itself 
therein,  if  the  condition  could  be  realized  which 
our  organs  of  sense  require. 

For  example  ;  Blakewell,  a  celebrated  English 
cattle-breeder,  observed,  in  a  great   number  of 


CHAP.  II.]  INDUCTION.  51 


individual  beasts,  a  tendency  to  fatten  readily,    Example  of 
and  in  a  great  number  of  others  the  absence  oi   ti^e  Engusb 


cattle 
breeder. 


this  constitution :  in  every  individual  of  the  for- 
mer description,  he  observed  a  certain  peculiar 
make,  though  they  differed  widely  in  size,  color, 
&:c.  Those  of  the  latter  description  differed  no 
less  in  various  points,  but  agreed  in  being  of  a 
different  make  from  the  others.  These /ac^5  were  How  he 
his  data;  from  which,  combining  them  with  the  the  facts: 
general  principle,  that  nature  is  steady  and  uni-     'Wiyhe 

off'  J  iuferred. 

form  in  her  proceedings,  he  logically/  drew  the 
conclusion  that  beasts  of  the  specified  make  have 
universally  a  peculiar  tendency  to  fattening. 

The  principal  difficulty  in  this  case  consisted  in  what  the 

diflScuIty 

in  making  the  observations,  and  so  collating  and    consisted. 
combining  them  as   to   abstract  from  each  of  a 
multitude  of  cases,  differing  widely  in  many  re- 
spects,   the    circumstances    in   which    they   all 
agreed.     But  neither  the  making  of  the  observa- 
tions, nor  their  combination,  nor  the  abstraction, 
nor  the  judgment  employed  in  these  processes, 
constituted  the  induction,  though  they  were  all 
preparatory  to  it.     The  Induction  consisted  in  in  what  the 
the  generalization ;  that  is,  in  inferring  from  all    consjgted. 
the  data,  that  certain  circumstances  would  be 
found  in  the  whole  class. 

The  mind  of  Newton  was  led  to  the  universal 
law,  that  all  bodies  attract  each  other  by  forces 


52 


LOGIC, 


[book  I. 


Newton's 

inference  of 

the  law  of 

universal 

gravitation. 


How  he 

observed 

facts  and 

their 

connections. 


The  use 
which  he 
made  of 
exact 
science. 


What  was 
the  result. 


varying  directly  as  their  masses,  and  inversely 
as  the  squares  of  their  distances,  by  Induction. 
He  saw  an  apple  falling  from  the  tree  :  a  mere 
fact ;  and  asked  himself  the  cause  ;  that  is,  if  any 
inference  could  be  drawn  from  that  fact,  which 
should  point  out  an  invariable  antecedent  condi- 
tion. This  led  him  to  note  other  facts,  to  prose- 
cute experiments,  to  observe  the  heavenly  bodies, 
until  fi'om  many  facts,  and  their  connections 
with  each  other,  he  arrived  at  the  conclusion, 
that  the  motions  of  the  heavenly  bodies  were  gov- 
erned by  general  laws,  applicable  to  all  matter , 
that  the  stone  whirled  in  the  sling  and  the  earth 
rolling  forward  through  space,  are  governed  in 
their  motions  by  one  and  the  same  law.  He 
then  brought  the  exact  sciences  to  his  aid,  and 
demonstrated  that  this  law  accounted  for  all  the 
phenomena,  and  harmonized  the  results  of  all  ob- 
servations. Thus,  it  was  ascertained  that  the 
laws  which  regulate  the  motions  of  the  heav- 
enly bodies,  as  they  circle  the  heavens,  also 
guide  the  feather,  as  it  is  wafted  along  on  the 
passing  breeze. 


The  ways  of 

ascertaining 

facts  are 

known: 


§  33.  We  have  already  indicated  the  ways  in 
which  the  facts  are  ascertained  from  which  the 
inferences  *  are  drawn.  But  when  an  inference 
can  be  drawn ;  how  many  facts  must  enter  into 


CHAP.  II.] 


INDUCTION, 


53 


the  premises ;  what  then'  exact  nature  must  be ; 
and  what  their  relations  to  each  other,  and  to 
the  inferences  which  flow  from  them ;  are  ques- 
tions which  do  not  admit  of  definite  answers. 
Although  no  general  law  has  yet  been  discov- 
ered connecting  all  facts  with  truths,  yet  all  the 
uniformities  which  exist  in  the  succession  of  phe- 
nomena, and  most  of  those  which  prevail  in  their 
coexistence,  are  either  themselves  laws  of  cau- 
sation or  consequences  resulting  and  corollaries 
capable  of  being  deduced  from,  such  laws.  It 
being  the  main  business  of  Induction  to  deter- 
mine the  effects  of  every  cause,  and  the  causes 
of  all  effects,  if  we  had  for  all  such  processes 
general  and  certain  laws,  we  could  determine, 
in  all  cases,  what  causes  are  correctly  assigned 
to  what  eflects,  and  v/hat  effects  to  what  causes, 
and  we  should  thus  be  virtually  acquainted  with 
the  whole  course  of  nature.  So  far,  then,  as  we 
can  trace,  with  certainty,  the  connection  be- 
tween cause  and  effect,  or  between  effects  and 
their  causes,  to  that  extent  Induction  is  a  sci- 
ence. When  this  cannot  be  done,  the  conclu- 
sions must  be,  to  some  extent,  conjectural. 


but  wo 

do  not  know 

certainly, 

in  all  cases, 

■when  we  can 

draw  on 

inference. 


No 
general  law. 


Business 

of 
Induction. 


What  is 
necessary. 


How  far  a 

science. 


54  LOGIC.  [book  I. 


CHAPTER    III. 

DEDUCTION — ^'A1^;RE    OF   THE    SYLLOGISM ITS    USES    AND    APPLICATIONS. 

DEDUCTION. 

§  34.    We    have    seen    that    all   processes    of 

Inductive     Reasoning,  in  which  the  premises  are  particular 

'!!!!!!!fnJ'    facts,    and    the    conclusions   general    truths,   are 

called  Inductions.     All  processes  of  Reasoning, 

in  which  the  premises  are  general  truths  and  the 

Deductive    conclusions  particular  truths,  are  called  Deduc- 

processes.    ^j^^^g^     Hcuce,    a   deduction   is    the    process    of 

Deduction    reasoning  by  which  a  particular  truth  is  inferred 

from  other  truths  which  are  known  or  admitted. 

The  formula  for  all  deductions  is  found  in  the 

Syllogism,  the  parts,  nature,  and  uses  of  which 

we  shall  now  proceed  to  explain. 


PROPOSITIONS. 

Proposition,       §  ^^-   ^  proposition  is  a  judgment  expressed 
ludgmentm  £^  wovds.     Hcnce,  a  proposition  is  defined  Jogi- 

words:  ^       '  " 

cally,    "  A    sentence    indicative :"    affirming   or 
*  Section  30. 


Deductive 
formula. 


CHAP.  III. 


PROPOSITIONS.  55 


denying;  therefore,   it  must   not  be  ambiguous,  must  not  be 

ambiguous; 

for  that  which  has    more  than  one  meanmg  is   norimper- 
in    reaUty  several    propositions ;    nor  imperfect,  ^fj^^^^^^ 
nor  ungrammatical,  for  such   expressions   have 
no  meaning  at  all. 


§  36.    Whatever  can  be  an  object  of  belief, 
or  even  of  disbelief,  must,  when  put  into  words,  a  proposition 

•   •  All  1  ]     explained. 

assume  the  lorm  oi  a  proposition.     All  truth  and 

all  error  lie   in  propositions.     What  we    call  a 

truth,  is   simply  a  true  proposition ;   and  errors  its  nature,- 

are  false  propositions.     To  know  the  import  of 

all  propositions,  would  be  to  know  all  questions 

which  can  be  raised,  and  all  matters  which  are  Embraces  aii 

truth  and  all 

susceptible   of  being   either   believed   or   disbe-       error. 

lieved.     Since,  then,  the  objects  of  all  belief  and 

all  inquiry  express  themselves  in  propositions,  a 

sufficient  scrutiny  of  propositions  and  their  va-  An  examina- 
tion of 
rieties  will  apprize  us  of  what  questions  mankind  propositions 

have  actually  asked  themselves,  and  what,  in  the  questtoMMU 


all  knowl- 

actually  thought  they  had  grounds  to  believe. 


nature  of  answers  to  those  questions,  they  have 


§  37.  The  first  glance  at  a  proposition  shows  Apropositioc 

,  .      .       f  ,    ,  .  ,  is  formed  b) 

that  it  is  lormed  by  putting  together  two  names,   p^ting  t„o 
Thus,  in  the  proposition,  "  Gold  is  yellow,"  the      "^"^^ 

^       ^  •'  together. 

property  yellow  is  affirmed  of  the  substance  gold. 
In   the  proposition,  "  Franklin  was    not  born  in 


56 


LOGIC. 


[book  I. 


England,"  the  fact  expressed  by  the  words  horn 
in  England  is  denied  of  the  man  Frankhn. 


A 
proposition 
has  three 

parts: 

Subject, 

Predicate, 

and 

Copula. 


Subject 
defined. 


Copula 
must  be 

IS   or  X3    NOT. 

All  verbs 

resolvable 

Into  "  to  be." 


§  38.  Every  proposition  consists  of  three 
parts  :  the  Subject,  the  Predicate,  and  the  Co- 
pula.  The  subject  is  the  name  denoting  the 
person  or  thing  of  which  something  is  affirmed 
or  denied :  the  predicate  is  that  which  is  affirm- 
ed or  denied  of  the  subject ;  and  these  two  are 
called  the  terms  (or  extremes),  because,  logically, 
the  subject  is  placed ^rs^,  and  the  predicate  last. 
The  copula,  in  the  middle,  indicates  the  act  ot 
judgment,  and  is  the  sign  denoting  that  there  is 
an  affirmation  or  denial.  Thus,  in  the  proposi- 
tion, "  The  earth  is  round ;"  the  subject  is  the 
words  "  the  earth,"  being  that  of  which  some- 
thing is  affirmed :  the  predicate,  is  the  word  round, 
which  denotes  the  quality  affirmed,  or  (as  the 
phrase  is)  predicated :  the  word  is,  which  serves 
as  a  connecting  mark  between  the  subject  and 
the  predicate,  to  show  that  one  of  them  is  af- 
firmed of  the  other,  is  called  the  Copula.  The 
copula  must  be  either  is,  or  is  not,  the  substan- 
tive verb  being  the  only  vej'h  recognised  by 
Logic.  All  other  verbs  are  resolvable,  by  means 
of  the  verb  "  to  be,"  and  a  participle  or  adjective. 
For  example : 

•'  The  Romans  conquered :" 


CHAP.  III.]  SYLLOGISM.  57 


the  word  "  conquered"  is  both  copula  and  predi-  Examples 

,     .                  .       ,                                      •    .       •          )}  of  the 

cate,  being  equivalent  to  "  were  victorious.  copuia. 
Hence,  we  might  write, 


"  The  Romans  were  victorious," 

in  which  were  is  the  copula,  and  victorious  the 
predicate. 


§  39.    A  proposition   being  a  portion    of  dis-  Apropositiou 

is  either 

course,  in  which  something  is  affirmed  or  denied  affirmative 
of  .something,,  all  propositions  may  be  divided 
into  affirmative  and  negative.  An  affirmative 
proposition  is  that  in  which  the  predicate  is  af- 
firmed of  the  subject ;  as,  "  Caesar  is  dead."  A 
negative  proposition  is  that  in  which  the  predicate 
is  denied  of  the  subject ;  as,  "  Caesar  is  not  dead." 
The  copula,   in  this  last  species  of  proposition,    in  the  last, 

f,      ,  ,  ,,,.,.,        the  copula  ia. 

consists   01    the   words   "  is   not,     which  is   the      jg  ^^^ 
sign  of  negation  ;   "  is"  being  the  sign  of  affirm- 
ation. 


SYLLOGISM. 

§  40.  A  syllogism  is  a  form  of  stating  the  con-   a  syllogism 

consists  of 

nection  which    may    exist,   for    the    purpose    of  three  propo 
reasoning,  between   three  propositions.      Hence, 
to    a   leffitimate    svllosism,    it    is    essential  that 

^  .        t>         '  Two  are 

there  should  be  three,  and  only  three,  proposi-    admitted: 


58  LOGIC.  [book  1. 

and  the  fhii'd  tioiis.     Of  these,   two  are   admitted  to  be    true, 
from  them.    ^^^^  ^^'^  Called  the  premises :  the  third  is  proved 
from  these  two,    and    is    called    the    conclusion. 
For  example  : 


Exomple. 


Rrajor  Term 


"  All  tyrants  are  detestable  : 
Caesar  was  a  tyrant ; 
Therefore,  Caesar  was  detestable." 

Now,  if  the  first  two  propositions  be  admitted, 

the  third,  or  conclusion,  necessarily  follows  from 

them,  and  it  is  proved  that  Caesar  was  detestable. 

Of  the  two  terms  of  the  conclusion,  the  Predi- 

dcfined,  ^^j-g  (detestable)  is  called  the  m.ajor  term,  and 
the  Subject  (Caesar)  the  ?ninor  term ;  and  these 
two  terms,  together  with  the  term  "  tyrant," 
make  up  the  three  propositions  of  the  syllogism, 
Minor  Term.  — each  term  being  used  twice.  Hence,  every 
syllogism  has  three,  and  only  three,  different 
terms. 

Major  The  premise,  into  which  the  Predicate  of  the 

Premise 

defined,     couclusion  eutcrs,  is  called  the  major  premise ; 

Minor      ^^^®  othcr  is  Called  the  minor  premise,  and  con- 

Premiae.     t^ius    the   Subjcct  of  the   conclusion ;    and  the 

other  term,  coinmon   to  the  two  premises,   and 

with  which  both  the  terms  of  the  conclusion  were 

separately  compared,  before  they  were  compared 

MiddieTenn.  -with  cach  Other,  is  Called  the  middle  term.     In 

the  syllogism  above,  "detestable"  (in  the  con- 


CHAP.  III.]  SYLLOGISM.  59 

elusion)  is  the  major  term,  and  "  Caesar"  the  mi-    Example, 

,  pointing  out 

nor  term :  hence,  i^jaj^j. 

premise, 

"  All  tyrants  are  detestable,  niin^r 

.  premise,  and 

IS  the  major  premise,  and  MiddieTorm. 

"  Caesar  was  a  tyrant," 
the  minor  premise,  and  "  tyrant"  the  middle  term. 

§  41.  The  syllogism,  therefore,  is  a  mere  for- 

^  J        &  '  '  Syllogism, 

mula  for  ascertaining  what  may,  or  what  may      a  mere 

formula. 

not,  be  predicated  of  a  subject.     It  accomplishes 

this  end  by  means   of  two  propositions,  viz.  by 

comparing  the    given   predicate   of  the   first   (a  Howappiied. 

Major  Premise),   and   the  given   subject  of  the 

second  (a  Minor  Premise),  respectively  with  one 

and  the  same  third  term  (called  the  middle  term), 

and  thus — under  certain  conditions,   or  laws  of 

the  syllogism — to  be  hereafter  stated — eliciting 

the  truth  (conclusion)   that  the  given  predicate 

must  be  predicated  of  that  subject.     It  will  be    use  of  the 

Major 

seen  that  the   Major  Premise  always   declares,     premise. 

in  a  general  way,  such  a  relation  between  the 

Major  Term  and  the  Middle  Term  ;  and  the  Mi-  of  the  Minor 

nor  Premise  declares,  in  a  more  particular  way, 

such   a  relation  between  the   Minor  Term  and 

the  Middle  Term,  as    that,  in   the   Conclusion,      ortho 

1  1        -HT    •        Middle  Tfc 

the  Minor  lerm  must  be  put  under  tlie  Major 
Term ;  or  in  other  words,  that  the  Major  Term 
must  be  predicated  of  the  Minor  Term. 


60  LOGIC.  [book  1. 


ANALYTICAL  OUTLINE  OF  DEDUCTION. 

Reasoning  §  42.  In  evGiy  instaiicG  in  which  we  reason, 
in  the  strict  sense  of  the  word,  that  is,  make  use 
of  arguments,  whether  for  the  sake  of  refuting 
an  adversary,  or  of  conveying  instruction,  or  of 
satisfying  our  own  minds  on  any  point,  whatever 
may  be  the  subject  we  are  engaged  on,  a  certain 
process  takes  place  in  the  mind,  which  is  one 

The  process,  and  the  same  in  all  cases  (provided  it  be  cor- 

thesame.  Tcctly  conductcd),"  whether  we  use  the  inductive 
process  or  the  deductive  formulas. 

Of  course  it  cannot  be  supposed   that  every 

Everyone    ouc  is  eveii  couscious  of  this  proccss  in  his  own 

not  conscious        ■      ■,  ii  •  i^x  i-^i 

of  the      mind;    much  less,   is   competent   to  explain  the 

process,     principles  on  which  it  proceeds.     This  indeed  is, 

The  same  for  and  cauuot  but  be,   the  case  with  every  other 

every  other  ,.  ,  .    ,  .  i  i 

process,  pi'occss  respecting  which  any  system  has  been 
formed  ;  the  practice  not  only  may  exist  inde- 
pendently of  the  theory,  but  must  have  preceded 
the  theory.     There  must  have  been  Language 

Eiementsimd  bcforc  a  systeiii  of  Grammar  could  be  devised ; 

knowledge  of  ^^^  musical  compositious,  previous  to   the   sci- 

elements,  '■  ^ 

must  precede  encc  of  Music.     This,  by  the  way,  serves  to  ex- 

gcneraliza- 

tion  and     pose  the  futility  of  the  popular  objection  against 
of  principles.  Logic  J  viz.  that  mcii  may  reason  very  well  who 
know  nothing  of  it.     The  parallel  instances  ad- 
duced show  that  such  an  objection  may  be  urged 


CHAP.  HI.]  ANALYTICAL     OUTLINE.  61 

in  many  other  cases,  where  its  absurdity  would  j^jgic 
be  obvious ;  and  that  there  is  no  ground  for  de- 
ciding thence,  either  that  the  system  has  no  ten- 
dency to  improve  practice,  or  that  even  if  it  had 
not,  it  might  not  still  be  a  dignified  and  inter- 
esting pursuit. 

§  43.    One   of  the   chief   impediments   to   the  lameness  of 

the  reasoning 

attainment  of  a  just  view  of  the  nature  and  ob-      process 
ject  of  Logic,  is  the  not  fully  understanding,  or  kept  in  mind, 
not  sufficiently  keeping  in  mind  the    sameness 
of  the  reasoning  process  in  all  cases.     If,  as  the 
ordinary  mode  of  speaking  would  seem  to  indi- 
cate,  mathematical   reasoning,    and    theological,  aii  kinds  of 

J  ,j         •!  1         Ti-ic  reasoning  are 

and  metaphysical,  and  pohtical,  otc,  were  essen-  alike  in 
tially  different  from  each  other,  that  is,  difierent  p"^<='p^« 
kinds  of  reasoning,  it  would  follow,  that  suppo- 
sing there  could  be  at  all  any  such  science  as 
we  have  described  Logic,  there  must  be  so  many 
different  species  or  at  least  different  branches 
of  Logic.  And  such  is  perhaps  the  most  pre- 
vailing notion.     Nor  is   this  much   to  be  won-    iJeasonof 

,1  .  .        .  .  ,  n        1  '^6  prevaili 

dered  at ;  since  it  is  evident  to  ail,  that  some 
men  converse  and  write,  in  an  argumentative 
way,  very  justly  on  one  subject,  and  very  erro- 
neously on  another,  in  which  again  others  excel, 
who  fail  in  the  former. 

This  error  may  be  at  once  illustrated  and  re- 


ing  errors 


62  LOGIC.  [book  I. 


The  reasonof  moved,  by  Considering  the  parallel  instance  of 
iuustrated  Arithmetic ;  in  which  every  one  is  aware  that 
by  example,  |.j^g  process  of  a  Calculation  is  not  affected  bv 

which  shows  '■  •' 

that  the  rea-  the  nature   of  the   objects  whose    numbers   are 

soniug 

process  is     before  US ;  but  that,   for  example,   the  multipli- 

always  the  ,  •  r  i  •         i 

gjjmg_       cation  01  a  number  is  the  very  same  operation, 
whether  it  be  a  number  of  men,  of  miles,  or  of 


pounds ;  though,  nevertheless,  persons  may  per- 
haps be  found  who  are  accurate  in  the  results 
of  their  calculations  relative  to  natural  philoso- 
phy, and  incorrect  in  those  of  political  econo- 
my, from  their  different  degrees  of  skill  in  the 
subjects  of  these  two  sciences ;  not  surely  be- 
cause there  are  different  arts  of  arithmetic  ap 
plicable  to  each  of  these  respectively. 

§  44.    Others  again,  who  are  aware  that  the 
s3oraeview    simple  systcm  of  Logic  may  be  applied  to  all 

Logic  as  a 

peciuiar     subjccts  whatcvcr,  are  yet  disposed  to  view  it 

method  of  i       i       /-  •  i 

reasoning:  ^s  a  peculiar  method  01  reasoning,  and  not,  as 
it  is,  a  method  of  unfolding  and  analyzing  our 
reasoning :  whence  many  have  been  led  to  talk 
of  comparing  Syllogistic  reasoning  with  Moral 
reasoning;  taking  it  for  granted  that  it  is  pos- 
sible to  reason  correctly  without  reasoning  logi- 

it  is  the  only  cally ;  which  is,  in  fact,  as  great  a  blunder  as  if 

method  of  .        ,  - 

reasoning     ^^7  ^ue  wcrc  to  mistake  grammar  lor  a  pecu- 
correctiy:    y\qx  language,  and  to  suppose  it  possible  to  speak 


CHAP.  III.]  ANALYTICAL     OUTLINE.  63 

correctly  without  speaking  grammatically.  They 
have,  in  short,  considered  Logic  as  an  art  of  rea- 
soning ;  whereas  (so  far  as  it  is  an  art)  it  is  the 
art  of  reasonina;:  the  logician's  obiect  beinsr,  not  RiaysdowR 

"=>'  =>  •'  ^  rules,  not 

to  lay  down  principles  by  which  one  may  reason,   which  maij, 

but  which 

but  by  which  all  must  reason,  even  though  they     must  be 
are  not  distinctly  aware  of  them : — to  lay  down 
rules,   not  which  may  be  followed  with   advan- 
tage,   but   which    cannot   possibly   be    departed 

from  in   sound  reasoning.     These  misapprehen-    Misappre- 
hensions and 
sions  and  objections  being  such  as  lie  on  the    objections 

very  threshold  of  the  subject,  it  would  have  been 
hardly  possible,  without  noticing  them,  to  con- 
vey any  just  notion  of  the  nature  and  design  of 
the  logical  system. 

§  45.    Supposing   it   then   to  have  been  per-  operation  of 

,      ,  ,  .  ~  ...  1,      reasoning 

ceived  that  the  operation  oi  reasonmg  is  m  all    should  be 
cases   the   same,   the  analysis  of  that  operation    '^'^^y^'"^^ 
could  not  fail  to  strike  the  mind  as  an  interesting 
matter  of  inquiry.     And  moreover,  since  (appa- 
rent) arguments,  which  are  unsound  and  incon- 
clusive, are  so  often  employed,  either  from  error  Because  such 

T      .  1       .  ,  ,  ,       analysis  is 

or  desiffn ;    and  since  even  those  who  are  not  „„  ,„ 

o     '  necessary  to 

misled  by  these  fallacies,  are  so  often  at  a  loss    furnish  the 
to  detect  and  expose    them  in  a  manner  satis- 
factory to  others,  or  even  to  themselves  ;  it  could 
not  but  appear  desirable  to  lay  down  some  gen- 


64  LOGIC.  [book  1, 

rules  for  the  crul  rulcs  of  reasoning,  applicable  to  all  cases ; 
ion  o    1     -which  a  person  misht  be  enabled  the  more 

error  and  the      -J  i  o 

discovery  of  readily  and   clearly  to  state  the  grounds   of  his 

truth, 

own  conviction,  or  of  his  objection  to  the  argu- 
ments of  an  opponent;  instead  of  arguing  at 
random,  without  any  fixed  and  acknowledged 
principles  to  guide  his  procedure.  Such  rules 
Such  rules    ^yQyifj  \yQ  analogous  to  those  of  Arithmetic,  which 

iu-e  analogous  " 

totheruiesof  obviatc  the  tediousness  and  uncertainty  of  cal- 

Arithmetic. 

culations  in  the  head  ;  wherein,  after  much  labor, 
different  persons  might  arrive  at  different  results, 
without  any  of  them  being  able  distinctly  to 
point  out  the  error  of  the  rest.  A  system  of 
such  rules,  it  is  obvious,  must,  instead  of  deserv- 
They  bring   iug  to  be  Called  the  art  of  wrangling,  be  more 

the  parties,  in    .         i  i  •        i  i  r  ■  i 

argument,  to  Justly  characterized  as  the  "art  of  cuttmg  short 
*"'^*"®"     wrangling,"  by  bringing  the  parties  to  issue  at 
once,  if  not  to   agreement;   and   thus   saving  a 
waste  of  ingenuity. 


Every  con-        §  4g    Jn  pursuing  the  supposed  investigation, 

elusion  is      _  , 

deducedfrom  it  will  be  found  that  in  all  deductive  processes 
two  proposi-  Qy^YY  conclusion  is  deduced,  in  reality,  from  two 

tions,  called  -  '' 

Premises,     other  propositious  (thence  called  Premises)  ;  for 
ifoneprem-  though  One  of  thcsc  may  be,  and  commonly  is, 

ise  is  sup-  "  ''  .   •' 

pressed,  it  is  supprcsscd,  it  must    nevertheless  be  understood 

nevertheless 

understood,    as  admitted ;  as  may  easily  be  made  evident  by 
supposing  the  denial  of  the  suppressed  premise; 


CHAP.   fTI.]  ANALYTICAL     OUTLINE.  65 

which  will  at  once  invalidate  the  argument.     For 
example ;  in  the  following  syllogism  : 

"  Whatever  exhibits  marks  of  design  had  an  intelligent  author: 
The  world  exhibits  marks  of  design  ; 
Therefore,  the  world  had  an  intelligent  author  :" 

if  any  one  from  perceiving  that  "  the  world  ex- 
hibits marks  of  design,"  infers  that  "it  must  have      and  is 
had  an  intelligent  author,"  though  he  may  not  be  "^'^'^^^^  *« 

°  '  &  J  the  argu- 

aware  in  his  own  mind  of  the  existence  of  any  ment,  though 

oue  may  no! 

other  premise,  he  will  readily  understand,  if  it  be    be  aware 
denied  that  "  whatever  exhibits  marks  of  design 
must  have  had  an  intelligent  author,"  that  the 
affirmative   of  that  proposition  is  necessary  to 
the  validitv  of  the  argument. 


§  47.  When  one  of  the  premises  is  suppressed  Enthymemc-. 

a  syllogism 

(which  for  brevity's  sake  it  usually  is),  the  argu-     with  one 
ment  is  called  an  Enthymeme.     For  example : 


of  it. 


premise 
suppressod- 


"  The  world  exhibits  marks  of  design, 

Tlierefore  the  world  had  an  intelligent  author," 

is  an  Enthymeme.     And  it  may  be  worth  while 

to  remark,  that,  when  the   argument  is  in  this    objections 

.  made  to  the 

state,  the  objections  of  an  opponent  are  (or  rather  assertion  or 
appear  to  be)  of  two  kinds,  viz.  either  objections  ^^^l^Zllt- 
to  the  assertion  itself,  or  objections  to  its  force      ™'^"'- 
as  an   argument.      For  example :   in   the   above    Exampiu 
instance,  an  atheist  may  be  conceived  either  de- 

5 


66  LOGIC.  [book  1. 


nyino;  that  the  world  does  exhibit  marks  of  de- 

Bolh  prein-        J       O 

ises  must  be  gigrn  Or  denying  that  it  follows  from  thence  that 

true,  if  the     /=  . 

argument  is   it  had  an  intelhgent  author.     Now  it  is  impor- 

sound : 

tant  to  keep  in  mind  that  the  only  difference  in 

the  two  cases  is,  that  in  the  one  the  expressed 

premise  is  denied,  in  the  other  the  suppressed ; 

and  when    for  the /orce  as  an  argument  of  either  premise 

theconciu-'  depends  on  the  other  premise  :  if  both  be  admit- 

sion  follows,  ^g^^  ^j^g   conclusion  legitimately  connected  with 

them  cannot  be  denied. 


§  48.  It  is  evidently  immaterial  to  the  argu- 
ment whether  the  conclusion  be  placed  first  or 
Premise     last ;    but   it  may  be  proper  to   remark,   that   a 
the  conciu-   premise  placed  after  its  conclusion  is  called  the 
Dion  is  called  ji^f^gQ^j  ^f  [^  ^ud  is  iutroduccd  by  one  of  those 

the  Reason.  •' 

conjunctions  which  are  called  causal,  viz.  "  since," 

"because,"  &c.,  which  may  indeed  be  employed 

to  designate  a  premise,  whether  it  come  first  or 

Illative      l^st.     The  iUativc  conjunctions  "  therefore,"  &c., 

conjunction,  ^^ggig^ate  the  couclusion. 

It  is   a  circumstance    which    often    occasions 
Causes  of    qytoy  and  perplexity,  that  both  these  classes  of 

error  and 

perplexity,  conjunctions  have  also  another  signification,  be- 
ing employed  to  denote,  respectively,  Cause  and 
Effect,  as  well  as  Premise  and  Conclusion.     For 

Different  "^ 

significations  example :  if  I  say,  "  this  ground  is  rich,  because 

of  the  .  n  •    ^   ■  ;> 

ccniunctions.  the  trccs  on  it  are  iiourishmg ;    or,  "  the  trees  are 


CHAP.   in. J  ANALYTICAL     OUTLINE.  67 

flourishing,  and  therefore  the  soil  must  be  rich ;"  Examples 
I  employ  these  conjunctions  to  denote  the  con-  conjimctiona 
nection  of  Premise   and    Conclusion ;    for  it   is     ^'^^  "^'"'^ 

logically. 

plain  that  the  luxuriance  of  the  trees  is  not  the 
cause  of  the  soil's  fertility,  but  only  the  cause 
of  my  knowing  it.  If  again  I  say,  "  the  trees 
flourish,  because  the  ground  is  rich ;"  or  "  the 
ground  is  rich,  and  therefore  the  trees  flourish,'      Examples 

T  •  1  ■  ,  •  X        1  J        where  they 

1  am  usmg  the  very  same  conjunctions  to  denote  denote  caust 
the  connection  of  cause  and  effect;  for  in  this    ^"<i''^«<='- 
case,  the  luxuriance  of  the  trees  being  evident 
to  the  eye,  would  hardly  need  to  be  proved,  but 
might  need  to  be  accounted  for.      There   are.  Many  cases 

1  •  1  •    1     ii  •  in  which  the 

however,  many  cases,  m  which  the  cause  is  em-    cause  and 
ployed  to  prove  the  existence  of  its  effect ;  espe-    *^'^  '''^^*''" 

•■      •/  J^  A         are  thesanMj, 

cially  in  arguments  relating  to  future  events;  as, 
for  example,  when  from  favorable  weather  any 
one  argues  that  the  crops  are  likely  to  be  abun- 
dant, the  cause  and  the  reason,  in  that  case,  co- 
incide ;  and  this  contributes  to  their  being  so 
often  confounded  together  in  other  cases. 

§  49.    In   an  argument,   such   as  the  example     in  every 
above  given,  it  is,  as  has  been  said,  impossible  ^^^menjto"*^ 
for  any  one,  who  admits  both  premises,  to  avoid    admit  the 

premise  is  tu 

admitting  the  conclusion.     But  there  will  be  fre-    admit  the 

.  .  conclusion. 

quently  an  apparent  connection  of  premises  with 
a   conclusion   which  does  not  in   reality  follow 


premises 
conclusion 
must  not  be 


68  LOGIC.  [book  I. 

Apparent  from  them,  though  to  the  inattentive  or  unskilful 
anj  the  argument  may  appear  to  be  valid  ;  and  there 
are  many  other  cases  in  which  a  doubt  may  exist 

relied  on.  whether  the  argument  be  valid  or  not ;  that  is, 
whether  it  be  possible  or  not  to  admit  the  prem- 
ises and  yet  deny  the  conclusion. 

General  rules      §  50.    It  is  of  the  highest  importance,  there- 
for argiunen-   r  j.       t  j  i         /•  x  i  •    u 

tatlon  i(>^^^>  to  lay  down  some  regular  lorm  to  which 
necessary,  every  valid  argument  may  be  reduced,  and  to 
devise  a  rule  which  shall  show  the  validity  of 
every  argument  in  that  form,  and  consequently 
the  unsoundness  of  any  apparent  argument  which 
cannot  be  reduced  to  it.  For  example ;  if  such 
an  argument  as  this  be  proposed : 

Example  of  "  Every  rational  agent  is  accountable  : 

an  imperfect  Brutes  are  not  rational  agents  ; 

argument. 

Therefore  they  are  not  accountable  , 

or  again : 

2d  Example.    "  AH  wise  legislators  suit  their  laws  to  the  genius  of  their 
nation  ; 
Solon  did  this;  therefore  he  was  a  wise  legislator  :" 

Difficulty  of  there  are  some,  perhaps,  who  would  not  per- 
detectmgthe  j,gj^g  ^^y.  f^Hacv  in  such  arguments,  especiallv 
if  enveloped  in  a  cloud  of  words ;  and  still  more, 
when  the  conclusion  is  true,  or  (which  comes  to 
the  same  point)  if  they  are  disposed  to  believe 
it ;  and  others  might  perceive  indeed,  but  might 


CHAP.  III.]  ANALYTICAL     OUTLINE.  69 


be  at  a  loss  to  explain,  the  fallacy.     Now  these     To  what 
(apparent)     arguments    exactly    correspond,    re-        renf'^'*' 

arguments 
correspond. 


spectiveiy,  with  the  following,  the  absurdity  of 
the  conclusions  from  which  is  manifest : 


"  Every  horse  is  an  animal :  A  similar 

Sheep  are  not  horses  ;  ^^'^'^P'^ 

Therefore,  they  are  not  animals." 


And 


"  All   vegetables  grow  ;  SM  similar 

An  animal  grows  ;  example. 

Therefore,  it  is  a  vegetable." 

These  last  examples,  I  have  said,  correspond    Tiieseiast 
exactly  (considered  as  arguments)  with  the  for-     ^ith^the 
mer ;  the  question  respecting  the  vahdity  of  an     former, 
argument  being,  not  whether  the  conclusion  be 
true,  hut  whether  it  follows  from  the  premises 
adduced.     This  mode  of  exposing  a  fallacy,  by  This  mode  o/ 
bringing  forward  a  similar  one  whose  conclusion  ,  ^^p"*'"^ 

o      o  fallacy  some- 

is  obviously   absurd,   is   often,  and  very  ad  van-       *™^^ 

resorted  to. 

tageously,  resorted  to  in  addressing  those  who 
are  ignorant  of  Logical  rules ;  but  to  lay  down 
such  rules,  and  employ  tham  as  a  test,  is  evi-  Toiaydown 
dently  a  safer  and  more  compendious,  as  well  best  way 
as  a  more  philosophical  mode  of  proceeding.  To 
attain  these,  it  would  plainly  be  necessary  to 
analyze  some  clear  and  valid  arguments,  and  to 
observe  in  what  their  conclusiveness  consists. 


70  LOGIC.  [book  I, 

§  51.  Let  us  suppose,  then,  such  an  examin- 
ation to  be  made  of  the  syllogism  above  men- 
tioned : 

Example  of   "  Whatever  exhibits  marks  of  design  had  an  intelligent  author; 
a  perfect        rpj^^  world  exhibits  marks  of  design  ; 

Therefore,  the  world  had  an  intelligent  author." 

What  is  In  the  first  of  these  premises  we  find  it  as- 
the  first  sumed  universally  of  the  class  of  "  things  which 
premise,  exhibit  marks  of  design,"  that  they  had  an  intel- 
in  the  second  ligent  author ;  and  in  the  other  premise,  "the 
world"  is  referred  to  that  class  as  comprehended 
What  we     jj^  {[ .  j-^qw  it  is  evident  that  whatever  is  said  of 

may  iufer. 

the  whole  of  a  class,  may  be  said  of  any  thing 
comprehended  in  that  class ;  so  that  we  are  thus 
authorized  to  say  of  the  world,  that  "  it  had  an 
intelligent  author." 
Syllogism  Again,   if  we   examine    a    syllogism    with   a 

with  a  . 

negative     negative  conclusion,  as,  lor  example, 

conclusion. 


"  Nothing  which  exhibits  marks  of  design  could  liave  been 
produced  by  chance  ; 
The  world  exhibits,  &c.  ; 

Therefore,  the  world  could  not  have  been   produced  by 
chance," 

The  process  the  proccss  of  reasoning  will  be  found  to  be  the 

of  reasoning 

the  same.  Same;  sincc  it  is  evident  that  whatever  is  denied 
universally  of  any  class  may  be  denied  of  any 
thing  that  is  comprehended  in  that  class. 


CHAP.  III.  I  ANALYTICAL     OUTLINE.  71 


§  52.  On  further  examination,  it  will  be  found     mi  valid 

arguments 

that  all  valid  arguments  whatever,  which  are  reducible  to 
based  on  admitted  premises,  may  be  easily  re-  '  ^^orm"^ 
duced  to  such  a  form  as  that  of  the  foregoing 
syllogisms ;  and  that  consequently  the  principle 
on  which  they  are  constructed  is  that  of  the  for- 
mula of  the  syllogism.  So  elliptical,  indeed,  is  the 
ordinary  mode  of  expression,  even  of  those  who     ordinary 

•'  ^  mode  of 

are  considered  as  prolix  writers,  that  is,  so  much   expressing 

arguments 

is  implied  and  left  to  be  understood  in  the  course  elliptical. 
of  argument,  in  comparison  of  what  is  actually 
stated  (most  men  being  impatient  even,  to  excess, 
of  any  appearance  of  unnecessary  and  tedious 
formality  of  statement),  that  a  single  sentence 
will  often  be  found,  though  perhaps  considered 
as  a  single  argument,  to  contain,  compressed 
into  a  short  compass,  a  chain  of  several  distinct 
arguments.     But  if  each  of  these  be  fully  devel-    ^"^  *''•'" 

o  •'  fully  devel- 

oped, and  the  whole  of  what  the  author  intended    oped,  they 

may  all  be 

to  imply  be  stated  expressly,  it  will  be  found  that  educed  into 
all  the  steps,  even  of  the  longest  and  most  com-    ^  f^^m""*' 
plex  train  of  reasoning,  may  be  reduced  into  the 
above  form. 

§  53.  It  is  a  mistake  to  imagine  that  Aristotle 
and  other  logicians  meant  to  propose  that  this    '''*°  "^ 

o  i       i  not  mean 

prolix  form  of  unfolding  arguments  should  uni-    that  every 

argument 

versally  supersede,  in  argumentative  discourses,    should  be 


72  LOGIC.  [book  1. 


thrown  into  the  common  forms  of  expression ;  and  that  "  to 
syUogLm.'^  reason  logically,"  means,  to  state  all  arguments 
at  full  length  in  the  syllogistic  form ;  and  Aris- 
totle has  even  been  charged  with  inconsistency 
for  not  doing  so.     It  has  been  said  that  he  "  ar- 
gues like  a  rational  creature,  and  never  attempts 
That  form  is  to  bring  his  own  system  into  practice."     As  well 
™o7tmth'^'  might  a  chemist  be  charged  with  inconsistency 
for  making  use  of  any  of  the    compound   sub- 
stances  that  ai'e    commonly  employed,   without 
previously    analyzing    and   resolving   them    into 
Analogy  to    their  simplc  elements;  as  well  might  it  be  im- 

the  chemist. 

agined  that,  to  speak  grammatically,  means,  to 
parse  every  sentence  we  utter.  The  chemist 
(to  pursue  the  illustration)  keeps  by  him  his  tests 
and  his  method  of  analysis,  to  be  employed  when 
The  analogy  any  substancc  is  offered  to  his  notice,  the  com- 
position of  which  has  not  been  ascertained,  ov 
in  which  adulteration  is  suspected.  Now  a  fal- 
To  what  a    lacy  may  aptly  be  compared  to  some  adulterated 

fallacy  may  ,  .  .  ^  .  .  .     , 

becomp:aod.  compouud ;  "it  cousists  01  an  ingenious  mixture 
of  tiuth  and  falsehood,  so  entangled,  so  intimate- 
ly blended,  that  the  falsehood  is  (in  the  chemical 
phrase)  held  in  solution  :  one  drop  of  sound  logic 

iiow  detect-  ^^  ^'^^^  ^^^^  which  immediately  disunites  them, 
""^  makes  the  foreign  substance  visible,  and  precipi- 
tates it  to  the  bottom." 


CHAP.  III.]  ANALYTICAL     OUTLINE.  73 


ARISTOTLES    DICTUM. 

§  54.  But  to  resume  the  investigation  of  the     Form  of 
principles  of  reasonino; :  the  maxim  resulting  from    ^""^^^  '^. 

i^  I  O  &  argument. 

the  examination  of  a  syllogism  in  the  foregoing 
form,  and  of  the  application  of  which,  every  valid 
deduction  is  in  reality  an  instance,  is  this  : 

"  That  whatever  is  predicated  (that  is,  affirmed    Aristotle's 
or  denied)  universalis/,  of  any  class   of  things, 
may  be  predicated,  in  like  manner  (viz.  affirmed 
or  denied),  of  any  thing  comprehended  in  that 
class." 

This  is  the  principle  commonly  called  the  die-    what  the 
tu?Ji    de    omni    et   nullo,    for    the    indication    of    i"''"'^'p'^ 

IS  called. 

which  we  are  indebted  to  Aristotle,  and  which 
is  the  keystone  of  his  whole  logical  system.  It 
is    remarkable    that    some,    otherwise   iudicious  „^  ,    ., 

'  J  vMiat  writers 

writers,   should   have   been  so  carried   away  by  h^^vesaidof 

this  princi- 

their  zeal   against  that  philosopher,  as  to  speak     pie;  and 

why. 

with  scorn  and  ridicule  of  this  principle,  on 
account     of     its    obviousness    and    simplicitv  ',  ^.    ,. . 

i  •'        Simplicity  a 

though   they  would    probably  perceive   at  once      *'^®'°'' 

science. 

in  any  otlier  case,  that  it  is  the  greatest  tri- 
umph of  philosophy  to  refer  many,-  and  seem- 
ingly very  various  phenomena  to  one,  or  a  very 
few,  simple  principles ;  and  that  the  more  simple 
and  evident  such  a  principle  is,  provided  it  be 
truly  apphcable  to  all  the  cases  in  question,  the 


74  LOGJC.  [book  I. 


No  solid  Ob-  greater  is  its  value   and  scientific  beauty.      If, 

jection  to  the   .      ,        -  .        .    ,      ,  ,     , 

principle     Indeed,  any  prniciple  be  regarded  as  not  thus  ap- 
ever  urged,   pjjcable,  that  is  an  objection  to  it  of  a  different 
kind.     Such  an  objection  against  Aristotle's  dic- 
tum, no  one  has  ever  attempted  to  establish  by 
been  taken    ^^^7  ^^^^^  ^^  proof ;  but  it  has  oftcn  been  taken 
for  granted.  Jqt  granted ;  it  being  (as  has  been  stated)  very 
syuogism    commonly  supposed,  without   examination,  that 

not  a  distinct 

kind  of  ar-  the  syllogism  is  a  distinct  kind  of  argument,  and 
^"Tform^"'  that  the  rules  of  it  accordingly  do  not  apply,  nor 
applicable  to  -were  intended  to  apply,  to  all  reasoning  what- 

3.J1  CflSCS* 

ever,  whei'e  the  premises  are  granted  or  known. 


Objection:        §  55.  One  objection  against  the  dictum  of  Aris- 
t  attiosyi-  ^Q^T^Q  j|.  jyiay  be  worth  while  to  notice  briefly,  for 

logisin  was  •'  •'  ' 

intended  to  i\-^q.  g^ke  of  Setting  in  a  clearer  light  the   real 

make  a  dem- 

onstration    character  and  object  of  that  principle.     The  ap- 

plainer: 

plication  of  the  principle  being,  as  has  been 
seen,  to  a  regular  and  conclusive  syllogism,  it 
has  been  urged  that  the  dictum  was  intended 
to  prove  and  make  evident  the  conclusiveness 
of  such  a  syllogism ;  and  that  it  is  unphilo- 
sophical  to  attempt  giving  a  demonstration  of 
a  demonstration.  And  certainly  the  charge 
to  increase   would   be  just,    if  wc    could  imagine    the  logi- 

tho  certainty       .        ,  ,  .  -.  , 

ofa        cian  s    object  to    be,    to    increase    the    certainty 


conclusion. 


of  a  conclusion,  which  we  are  supposed  to  have 
already  arrived  at  by  the  clearest  possible  mode 


CHAP.  III.]  ANALYTICAL     OUTLINE.  75 

of  proof.     But  it  is  very  strange  that  such  an  This  view  u 

111        entirely 

idea  should  ever  have  occurred  to  one  who  had  enoneous. 
even  the  shghtest  tincture  of  natural  philosophy  ; 
for  it  might  as  well  be  imagined  that  a  natural  illustration. 
philosopher's  or  a  chemist's  design  is  to  strength- 
en the  testimony  of  our  senses  by  a  priori  rea- 
soning, and  to  convince  us  that  a  stone  when 
thrown  will  fall  to  the  ground,  and  that  gunpow- 
der will  explode  when  fired ;  because  they  show 
according  to  their  principles  those  phenom.ena 
must  take  place  as  they  do.  But  it  v/ould  be 
reckoned  a  mark   of  the   grossest  ignorance  and 

,,.,..     The  object  ia 

stupidity   not  to  be  aware  that   tnen*    object    is  not  to  prove, 
not    to    prove    the    existence   of   an    individual      "  °^^' 

J  count  for 

phenomenon,  which  our  eyes  have  witnessed, 
but  (as  the  phrase  is)  to  account  for  it ;  that  is, 
to  show  according  to  what  principle  it  takes 
place  ;  to  refer,  in  short,  the  individual  case  to 
a  ffeneral  laio  of  nature.     The  object  of  Aris-  I'^e  object  of 

°  -^  the  Dictum 

totle's   dictum  is  precisely   analogous:    he    had,   to  point  out 

the  general 

doubtless,  no  thought  of  adding  to  the  force  of  process  \o 

,-     •  T       1         11       •  1  •       1       •  X  -J.     which  each 

any  individual  syllogism ;  his  design  was  to  point 


case  con- 
forms. 


out  the  general  -principle  on  which  that  process 
is  conducted  which  takes  place  in  each  syllo- 
gism.    And  as  the  Laws  of  nature  (as  they  are     Laws  of 

nature,  gen- 

called)  are  in  reality  merely  generalized  facts,  of  eraiized  facta 
which  all  the  phenomena  coming  under  them  are 
particular  instances ;  so,  the  proof  drawn  from 


76  LOGIC.  [book  1 

The  Dictum  Aristotle's  dictum  is  not  a  distinct  demonstration 
form  of  all  brought  to  confirm  another  demonstration,  but  is 
demonsLra-   j^g;^-g]y  ^  generalized  and  abstract  statement  of 

(ion.  "^  " 

all  demonstration  whatever ;  and  is,  therefore,  in 
fact,  the  very  demonstration  which,  under  proper 
suppositions,  accommodates  itself  to  the  various 
subject-matters,  and  which  is  actually  employed 
in  each  particular  case. 


How  to  trace       §  56.    In   oi'dcr  to  trace  more  distinctly  the 

the  abstract-  r       i  i  •  i 

ingand      diiierent    steps    ot    the    abstracting   process,    oy 
reasoning    -^y^ich  any  particular  argument  may  be  brought 

process.  ./    r  c  J  o 

into  the  most  general  form,  we  may  first  take  a 

syllogism,  that  is,  an  argument  stated  accurately 

Anarg-ument  and  at  fuU  length,  such  as  the  example  formerly 

stated  at  full 

length,      given : 

"  Whatever  exhibits  marks  of  design  had  an  intelligent  author; 
The  world  exhibits  marks  of  design  ; 
Therefore,  the  world  had  an  intelligent  author :" 

Propositions  ^^^  thcii  somcwhat  generalize  the  expression,  by 
expressed  by  substituting   (as  in   Algebra)  arbitrary  unmean- 

abstract  &     \  O  /  .; 

terms.  jjjg  symbols  for  the  significant  terms  that  were 
originally  used.  The  syllogism  will  then  stand 
thus  : 

"  Every  B  is  A ;  C  is  B  ;  therefore  C  is  A." 

Tiie  reason-        The  reasoning,  when  thus  stated,  is  no  less  evi- 
"vaiirt,       dently  valid,  whatever  terms  A,  B,  and  C  respect- 


CHAP.   III.]  ANALYTICAL     OUTLINE.  77 


ively  may  be  supposed  to  stand  for ;  such  terms  and 
may  indeed  be  inserted  as  to  make  all  or  some  general. 
of  the  assertions  false ;  but  it  will  still  be  no  less 
impossible  for  any  one  who  admits  the  truth  of 
the  premises,  in  an  argument  thus  constructed, 
to  deny  the  conclusion ;  and  this  it  is  that  con- 
stitutes the  conclusiveness  of  an  argument. 

Viewing,  then,   the   syllogism   thus   expressed,  syiiogismso 

viewed, 

it  appears  clearly  that  "  A  stands  for  any  thing  affirms  gen- 
whatevcr  that  is  affirmed  of  a  certain  entire  class"  jjetween  the 
(viz.  of  every  B),  "which  class  comprehends  or      '^™^" 
contains  in  it  something  else"  viz.  C  (of  which  B 
is,  in   the  second    premiss,   affirmed)  ;    and  that, 
consequently,  the  first  term  (A)  is,  in  the  conclu- 
sion, predicated  of  the  third  (C). 

§  57.  Now,  to  assert  the  validity  of  this  pro-  Another  form 

,      -  .  ,  , .  of  stating  the 

cess  now  beiore  us,  is  to  state  the  very  dictum     dictum, 
we   are   treating  of,  with  hardly  even  a  verbal 
alteration,  viz.  : 

1.  Any  thing  whatever,  predicated  of  a  whole    The  three 

.  things 

Class  ,  implied. 

2.  Under  which  class  something  else  is  con- 
tained ; 

3.  May  be  predicated  of  that  which  is  so  con- 
tained. Thesethree 

members 

The  three  members  into  which  the  maxim  is  correspond  to 

the  three 

here  distributed,  correspond  to  the  three  propo-  propositions 


78  LOGIC.  [bock  I. 

sitions   of  the  syllogism  to   which  they  are   in- 
tended respectively  to  apply. 
Advantage  of      The   advantage  of  substituting  for  the  terms, 
^arburar"^  in  a  regular  syllogism,  arbitrar}^,  unmeaning  sym- 
symboisfor  j^qJ^^  ^^q)^  g^g  letters  of  the  alphabet,  is  much  the 

the  terms. 

same  as  in  geometry  :  the  reasoning  itself  is  then 
considered,  by  itself,  clearly,  and  without  any 
risk  of  our  being  misled  by  the  truth  or  falsity 
of  the  conclusion ;  which  is,  in  fact,  accidental 
and  variable ;  the  essential  point  being,  as  far  as 
Connection,   ^^^  argument  is   concerned,   the  connection  be- 

the  essential 

point  of  the  ^106671  the  prcmiscs  and  the  conclusion.     We  are 

argument. 

thus  enabled  to  embrace  the  general  principle  of 
deductive  reasoning,  and  to  perceive  its  applica- 
bility to  an  indefinite  number  of  individual  cases. 
That  Aristotle,  therefore,  should  have  been  ac- 
Aristotie     cuscd  of  making  use  of  these  symbols  for  the 

right  in  using 

these  sym-  purposc  of  darkening  his  demonstrations,  and 
that  too  by  persons  not  unacquainted  with  geom- 
etry and  algebra,  is  truly  astonishing. 

Syllogism        §  58.  It  bclongs,  then,  exclusively  to  a  syllo- 

cqually  true  . 

whcnab-     gism,  propcrly  so  called   (that  is,  a  valid  argu- 
erm    j^gj-^|  g^  grated  that  its  conclusiveness  is  evidtmt 

are  used.  ' 

from  the  mere  form  of  the  expression),  that  if 
letters,  or  any  other  unmeaning  symbols,  be  sub- 
stituted for  the  several  terms,  the  validity  of  the 
argument  shall  still  be  evident.     Whenever  this 


CHAP.   III.]  ANALYTICAL     OUTLINE.  79 


is  not  the  case,  the  supposed  argument  is  either  whennotso, 

11  1  •    ,  ■       1  1  1  1  1   the  supposed 

unsound  and  sophistical,  or  else  may  be  reduced    arc-ument 
(without  any  alteration  of  its  meaning)  into  the  ^^^^"'^'^' 
syllogistic   form ;    in  which  form,   the   test   just 
mentioned  may  be  applied  to  it. 


§  59.  What  is  called  an  unsound  or  fallacious  Definition  of 

,  .  an  unsound 

argument,  that  is,  an  apparent  argument,  which    argument. 
is,  in  reality,  none,  cannot,  of  course,  be  reduced 
into  this  form ;  but  when  stated  in  the  form  most 
nearly  approaching  to   this    that  is  possible,  its    Whenre- 

,  .  .  ,  ^  duced  to  the 

lallaciousness  becomes  more  evident,  from  its  fonn,  the  m- 
nonconformity  to  the  foregoing  rule.  For  ex-  '^Xidenr*^ 
ample  : 

"  Whoever  is  capable  of  deliberate  crime  is  responsible  ;  Example. 

An  infant  is  not  capable  of  deliberate  crime  ; 
Therefore,  an  infant  is  not  responsible." 

Here  the  term  "responsible"  is  affirmed  uni-    Anaiysisof 
versally  of  "  those  capable  of  deliberate  crime  ;"    ^^*^  °^'^^"' 
it  might,  therefore,  according  to  Aristotle's  dic- 
tum, have  been  affirmed  of  any  thing  contained 
under  that  class ;  but,  in  the  instance  before  us, 
nothing  is   mentioned   as   contained  under  that  its  defective 
class ;  only,  the  term  "  infant"  is  excluded  from  "''^'^^^ut "' 
that  class ;    and    though  what  is   affirm^ed   of  a 
whole  class  may  be  affirmed  of  any  thing  that 
is  contained  under  it,  there  is  no  ground  for  sup- 
posing that  it  may  be  denied  of  whatever  is  not 


80  LOGIC.  [book  1. 


so  contained ;  for  it  is  evidently  possible  that  it 

the  argument  '^^J  ^^  applicable  to  a  whole  class  and  to  some- 

13  not  good,  thing  else  besides.     To  say,  for  example,  that  all 

trees  are  vegetables,  does  not  imply  that  nothing 

else  is  a  vegetable.     Nor,  when  it  is  said,  that 

What  the    j^jj  ^^,]^q  j^^.g  capable  of  deliberate  crime  are  re- 

gtateraent 

impbes.     sponsible,   does    this    imply  that    no   others    are 

responsible ;  for  though  this  may  be  very  true, 

What  is  to    it  has  not  been  asserted  in  the  premise  before  us  ; 

be  done  in  i     •  i  ^       •  c 

the  analysis  and  ui  the  analysis  oi  an  argument,  we  are  to 
''^''"  ,     discard  all  consideration  of  what  misht  be  as- 

argumeut.  ° 

serted ;  contemplating  only  what  actually  is  laid 

down  in  the  premises.     It  is  evident,  therefore, 

i-heone     that  such   an   apparent  argument  as  the   above 

abovedidnot  j^^^  ^^^^  comply  with   the  rule  laid  down,  nor 

comply  with  r  j 

uieruie.     g^n  be  SO  Stated  as  to  comply  v/ith  it,  and  is 
consequently  invalid. 


§  GO.  Again,  in  this  instance  : 

^jjothe,  "  Food  is  necessary  to  life  ; 

example.  Corn  is  food  ; 

Therefore  corn  is  necessary  to  life  :" 

In  what  the  t^g  term  "  ncccssary  to  life"  is  affirmed  of  food, 

argument  is 

defective,  but  uot  universally ;  for  it  is  not  said  of  every 
hind  of  food  +he  meaning  of  the  assertion  be- 
ing manifestly  that  some  food  is  necessary  to 
life  :  here  again,  therefore,  the  rule  has  not  been 
complied  with,  since  that  which  has  been  predi- 


CHAP.  III.]  ANALYTICAL     OUTLINE.  81 


Gated   (that  is,   affirmed  or  denied),   not  of  the     why  we 
whole,  but  of  apart  only  of  a  cerlain  class,  can-   cateor^com 
not  be,  on  that  ground,  predicated  of  whatever    ^''"'  ^^  , 

o  '    I  predicated  of 

is  contained  under  that  class.  fo"''- 


DISTRIBUTION  AND  NON-DISTRIBUTION  OF  TKRMS. 

§  61.  The  fallacy  in  this  last  case  is,  what  is  Fallacy  in  the 

last  example. 

usually  described  in  logical  language  as  consist- 
ing in  the  "  non-distribution  of  the  middle  term  ;"  Non-distribu- 

tioii  of  the 

that  is,  its  not  being  employed  to  denote  all  the  middle  term, 
objects  to  which  it  is  applicable.      In  order  to 
understand  this  phrase,  it  is  necessary  to  observe, 
that  a  term  is  said  to  be  "  distributed,"  when  it  is 
taken   universally,  that  is,  so  as  to  stand  for  all 
its    significates ;    and   consequently  "undistribu- 
ted," when  it  stands  for  only  a  portion  of  its  sig- 
nificates.*    Thus,  "all  food,"   or  every  kind  of  what d/stri^ 
tood,  are  expressions  which  imply  the  distribu- 
tion   of    the     term     "  food  ;"     "  some     food"    would  Non-distribu- 
tion. 

imply  its  non-distribution. 

Now,  it  is  plain,  that  if  in  each  premiss  a  part 
only  of  the  middle  term  is  employed,  that  is,  if 
it  be  not  at  all  distributed,  no   conclusion   can 

How  the  ex- 
be  drawn.     Hence,  if  in  the  example  formerly  ample  might 

adduced,  it  had  been  merely  stated  that  "  some-      ^^^.^^^ 


*  Section  15. 
6 


82  LOGIC.  [book  I. 

thing"    (not    "  whatever,"    or    "  every    thing") 

"  which  exhibits  marks  of  design,  is  the  work  of 

an    intelhgent    author,"    it  would  not   have   fol- 

whiitit      lowed,  from  the  world's  exhibitino;  marks  of  de- 

would  tlion  " 

haveiiDpiied.  sign,  that  that  is  the  work  of  an  intelligent  author 


Words  mark-       §  62.  It  is  to  be  obscrvcd  also,  that  the  words 

ingdislribu-  )>        i  •    i  11  i-        -i        • 

tionornon-  "^11    and  "  every,    which  mark  the  distribution 
not"iway"    ^^  ^  term,  and  "some,"  which  marks   its  non- 
expressed.    distribution,  are  not  always  expressed :  they  are 
frequently  understood,  and  left  to  be  supplied  by 
the    context ;    as,  for  example,   "  food   is   neces- 
sary ;"  viz.  "  some  food  ;"  "  man  is  mortal ;"  viz. 
Such  propo  "  every  man."     Propositions  thus  expressed  are 

sitious  ai'e 

cniied       called  by  logicians  " indefinite"  because  it  is  left 

Indefinite.  ,  •         1      i  i  f  r        ^ 

undetermined    by   the    lorm    oi    the    expression 
whether  the  subject  be  distributed  or  not.     Nev- 
ertheless  it   is    plain    that  in  every   proposition 
the  subject  either  is  or  is  not  meant  to  be  dis- 
tributed,   though    it    be    not    declared    whether 
But  every    it  is  or    iiot ;    Consequently,    every   proposition, 
must  be      whether  expressed    indefinitely  or  not,   must  be 
either       uudcrstood    as  either  "universal"  or    "particu- 

Universal  or 

partir-uiar.    lar ;"  thosc  being  called  universal,  in  which  the 

predicate    is   said  of  the  whole    of  the   subject 

(or,   in   other  words,   where    all   the  significates 

are   included) ;    and    those    particular,  in  which 

each.       only  a  part  of  them  is  included.     For  example  : 


CHAP.  III.]  ANALYTICAL     OUTLINE.  83 


"  All  men  are  sinful,"  is  universal  :   "  some  men  riiis  division 
are  sinful,"  particular;  and  this  division  of  prop-        /   " 
ositions,  having  reference  to  the  distribution  of 
the  subject,  is,  in  logical  language,  said  to  be  ac- 
cording to  their  "  quantUy." 


§  63.  But  the  distribution  or  non-distribution  Wsfribution 

of  lilt)  predi- 

of  the  predicate  is  entirely  independent  of  the  catehusno 

ryfertjncG  lo 

quantity  of  the  proposition ;    nor  are   the  signs     quantity. 
"  all"  and  "  some"  ever  affixed  to  the  predicate  ; 
because    its    distribution   depends    upon,   and    is  "'is'"«''erenco 

to  quality. 

indicated  by,  the  "  quality'  of  the  proposition  ; 
that  is,  its  being  affirmative  or  negative ;  it  being 
a  universal  rule,  that  the  predicate  of  a  negative 
proposition  is  distributed,  and  of  an  affirmative,   when  it  is 

^       '■  distributed ; 

undistributed.     The  reason    of  this   may   easily 

be  understood,  by  considerina;  that  a  term  which   ^he  reason 

•^  ^  _  of  this. 

stands  for  a  whole  class  may  be  applied  to  (that 
is,  affirmed  of)  any  thing  that  is  comprehended 
under  that  class,  though  the  term  of  which  it  is  xherredicate 

.^  rr  ^  T  r  I  a.       j.   of aflintialive 

thus  affirmed  may  be  ot  much  narrower  extent  j,.op„siaons 
than  that  other,  and  may  therefore  be  far  from   ""''y  ^'^  "P" 

•^  jjlicablo  to 

coinciding  with  the  whole  of  it.     Thus  it  may   tiie  subject, 
be  said  with  truth,  that  "the  Negroes  are  unciv-  much  wider 
ilized,"  though  the  term  "  uncivilized"  be  of  much      ^''*^" ' 
wider  extent  than   "  Negroes,"    comprehending, 
besides    them,    Patagonians,    Esquimaux,    &c. ; 
so  that  it  would  not  be  allowable  to  assert,  that 


84  LOGIC.  [book  r. 


Hence,  oniya  all  who  are  Uncivilized  are  Negroes."     It  is  ev- 
terra  is  used.  i^G'^t,    therefore,    that  it  is   a  pa?^t  only  of  the 
term    "uncivilized"   that    has  been    affirmed  of 
"  Negroes ;"    and  the   same  reasoning  applies  to 
every  affirmative  proposition. 
But  It  may        It    may   indeed    so   happen,    that   the    subject 
exiLtHh    ^^^   predicate    coincide,   that   is,    are    of   equal 
the  subject:  g;xtent ;   as,  for  example:   "all   men  are  rational 
animals  ;"  "  all  equilateral  triangles  are  equian- 
gular ;"  (it  being  equally  true,  that  "  all  rational 
this  not  im-    animals  are  men,"  and  that  "all  equiangular  tri- 

plied  in  the  ,  .,  i    jjv  i  •       •  •         ?•     j 

foi-mofthe    angles  are  equilateral ;  )  yet  this  is  not  implied 
expression.    ^^  ^j^^  form  of  the  expressiou ;    since  it  would 
be  no  less  true  that  "  all  men  are  rational  ani- 
mals," even  if  there  were  other  rational  animals 
besides  men. 
If  any  part  of      It  is  plain,  therefore,  that  if  any  part  of  the 
u^lp'^piiclwr  predicate  is  applicable  to  the  subject,  it  may  be 
to  the  sub-    affirmed,  and  of  course  cannot  be  denied,  of  that 

Joct,  it  may 

be  affirmed    subjcct ;   and   Consequently,  whcu  the   predicate 

of  the  sub-  ,  .       .         , . 

ject.        is    denied   of  the  subject,   this  implies   that    no 
part  of  that  predicate  is  applicable  to  that  sub- 
ject ;  that  is,  that  the  whole  of  the  predicate  is 
Ka predicate  denied  of  the  subject:  for  to  say,  for  example, 
Li^ecl  "he^  that  "  no  beasts  of  prey  ruminate,"  implies  that 
whole  predi-  jjgag^g  ^f  pj^.gy  ^re  excluded  from  the  whole  class 

cate  is  i       •! 

denied  of    of  ruminant  animals,  and  consequently  that  "  no 

the  subject.  .  .  „  *      i 

rummant  animals  are    beasts   oi    prey.        And 


CHAP.   III. J  ANALYTICAL     OUTLINE.  85 


hence  results  the  above-mentioned  rule,  that  the   Distribution 

,.        .,        .  r-     1  T  •       •         T     1    •  of  predicate 

distribution  oi  the  predicate  is  impued  in  nega-    i„,piit,jii, 
tive  propositions,  and  its  non-distribution  in  af-     "«^s"tive 

'■       '■  propositions: 

firmativeS.  non-Uistribu- 

tion  in 
aflirmatives. 

§  64.  It  is  to  be  remembered,  therefore,  that  Not  sufficient 

for  the  mid- 
it  is  not  sufficient  for  the  middle  term  to  occur  die  term  to 

,  .    .  .  •  r     1  occur  in  a 

in  a  universal  proposition ;  since  it  that  propo-    universal 

sition  be  an  affirmative,  and  the  middle  term  be  p'"'^!'"*'"'"'- 

the  predicate  of   it,  it  will    not  be   distributed. 

For  example :  if  in  the  example  formerly  given, 

it  had  been  merely  asserted,  that  "  all  the  works 

of  an  intelligent  author  show  marks  of  design," 

and  that  "  the  universe  shows  marks  of  design,"  u  must  be  so 

nothing  could  have  been  proved ;  since,  though    *^^Xthe 

both  these  propositions  are  universal,  the  middle  *^'''"*  °^  '^® 

conclusion, 

term  is  made  the  predicate  in  each,  and  both  are    that  those 

terras  may  be 

affirmative ;  and  accordingly,  the  rule  of  Aris-  compared  to- 
totle  is  not  here  complied  with,  since  the  term  ^^  '^'^' 
"  work  of  an  intelligent  author,"  which  is  to  be 
proved  applicable  to  "  the  universe,"  would  not 
have  been  affirmed  of  the  middle  term  ("  what 
shows  marks  of  design")  under  which  "  universe" 
is  contained ;  but  the  middle  term,  on  the  con- 
trary, would  have  been  affirmed  of  it. 

If,  however,  one  of  the  premises  be  negative,  if  oneprem- 
the  middle  term  may  then  be  made  the  predicate  '^^  ^^^s» 


86  LOGIC.  [book  I. 


live,  the  mid-  of  that,   and  will  thus,   according  to  the   above 
bemadJihl  remark,  be  distributed.     For  example : 

predicate  of 

that,  and  will  ,,  ,t  •         .        •       i  i      • 

,     .....  "  ]\o  ruminant  animals  are  predacious  : 

be  distnb-  '^ 

uted.  The  lion  is  predacious ; 

Therefore  the  lion  is  not  ruminant ;" 

this  is   a  valid  syllogism  ;  and   the   middle  term 

(predacious)   is  distributed   by   being   made    the 

The  form  of  predicate  of  a  negative  proposition.     The  form, 

thissyiio-    jj;^(jgg(j    Qf  the  syllogism  is   not   that  prescribed 

gism  will  not  ''        °  '■ 

beiiiatpre-  by  the  dictum  of  Aristotle,  but  it  may  easily  be 

scribed  by 

the  dictum,    reduced  to  that  form,  by  stating  the  first  prop- 
but  inny  be  .   .  ,  _ ,  ,       .  .        , 

reduced  to  it.  osition  thus  :  "  J\o  prcdacious  animals  are  ru- 
minant;" which  is  manifestly  implied  (as  was 
above  remarked)  in  the  assertion  that  "no  ru- 
minant animals  are  predacious."  The  syllogism 
will  thus  appear  in  the  form  to  which  the  dictum 
applies. 

AM  argil-  §65.  It  is  not  every  argument,  indeed,  that 
°i7rldi|™d*  can  be  reduced  to  this  form  by  so  short  and  sim- 
bysoshorta    |g  an  alteration  as   in  the  case  before  us.     A 

process.       -i 

longer  and  more  complex  process  will  often  be 
required,  and  rules  may  be  laid  down  to  facilitate 
this  process  in  certain  cases ;  but  there  is  no 
sound  argument  but  what  can  be  reduced  into 
But  all  argu-  this  form,  without  at  all  departing  from  the  real 
meutsmay   ^g^uing  and  drift  of  it;  and  the  form  will  be 


CHAP.   IK.]  ANALYTICAL     OUTLINE.  87 


found   (though  more  prolix  than  is  needed  for   be  reduced 
ordinary  use)  the  most  per 
argument  can  be  exhibited. 


, .  ^     ,  .  .  ,  .    ,  to  the  pre- 

ordmary  use)  the  most  perspicuous  m  which  an  gcribedform. 


§  66.  All  deductive  reasoning  whatever,  then,  AUdeauctivc 
rests  on  the  one  simple  principle  laid  down  by   rests  on  the 
Aristotle,  that  ^^'=''^'"- 

"  What  is  predicated,  either  affirmatively  or 
negatively,  of  a  term  distributed,  may  be  predi- 
cated in  like  manner  (that  is,  affirmatively  or  neg- 
atively) of  any  thing  contained  under  that  term." 

So  that,  when  our  object  is  to  prove  any  prop-  what  are  the 
osition,  that  is,  to  show  that  one  term  may  rightly  p™"^^^^" 
be  affirmed  or  denied  of  another,  the  process 
which  really  takes  place  in  our  minds  is,  that  we 
refer  that  term  (of  which  the  other  is  to  be  thus 
predicated)  to  some  class  (that  is,  middle  term) 
of  which  that  other  may  be  affirmed,  or  denied, 
as  the  case  may  be.     Whatever  the  subject-mat-   Thereasoii- 

ing  always 

ter  of  an  argument  may  be,  the  reasoning  itself,     the  same. 
considered  by  itself,  is  in  every  case  the  same 
process;    and  if  the  writers   against  Logic  had    Mistakes  of 
kept  this  in  mmd,  they  would  have  been  cautious      Logic. 
of  expressing  their  contempt  of  what  they  call 
"  syllogistic  reasoning,"  which  embraces  all  de- 
ductive reasoning;  and  instead  of  ridiculing  Aris- 
totle's principle  for  its  obviousness  and  simplicity,     Anstotie-s 
would  have  perceived  that  these  are,  in  fact,  its     ''""'"''® 


88  LOGIC.  [book  I. 


simple  and  highest  praise:  the  easiest,  shortest,  and  most 
evident  theory,  provided  it  answer  the  purpose 
of  explanation,  being  ever  the  best. 


RULES    FOR   EXAMINING    SYLLOGISMS. 

rests  of  the       §  67.  The   following  axioms  or  canons  serve 

validity  of 

syllogisms,  ss  tests  of  the  Validity  of  that  class  of  syllo- 
gisms which  we  have  considered. 

1st  test.  1st.  If  two  terms  agree  with  one  and  the  same- 

third,  they  agree  with  each  other. 

ad  test.  2d.  If  one  term  agrees  and  another  disagrees 

with  one  and  the  same  third,  these  two  disagree 
with  each  other. 
The  first  the       On  the  former  of  these  canons  rests  the  va- 

teat  of  all         _ 

affirmative    Hdity  of  affirmative  conclusions  ;  on  the  latter, 

conclusions.        f.  ,  .  c  1 1        •  ^  r      ^ , 

The  second  ^^   negative :    lor,    no    syllogism    can    be    laulty 

of  negative,  ^y^jj^j^  ^Qgg  j-^qj-  yjolate  these  canons ;  none  cor- 
rect which  does ;  hence,  on  these  two  canons 
are  built  the  following  rules  or  cautions,  which 
are  to  be  observed  with  respect  to  syllogisms, 
for  the  purpose  of  ascertaining  whether  those 
canons  have  been  strictly  observed  or  not. 

Every  syiio-  1  st.  Every  syllogism  has  three  and  only  three 
three  and    t^^^s ;  viz.  the  middle  term  and  the  two  terms 

only  three     q|-  ^j^g  Couclusion  t  the  tcmis  of  the  Conclusion 

terms. 

are  sometimes  called  extremes. 
Every  syiio-       2d.  Evcry  syllogism  h  is  three  and  only  three 


CHAP.  III.] 


ANALYTICAL     OUTLINE, 


89 


■propositions;  viz.  the  major  premise ;  the  minor     gismhas 

■,     ,  ,       .  three  and 

premise  ;  and  the  conclusion.  ^niy  three 

3d.    If  the  middle    term  is   ambiguous,    there  P™P"«'tions. 

.  Middle  term 

are  in  reality  two  middle  terms,  in  sense,  though  must  not  bo 
but  one  in  sound.  ambiguous. 

There  are  two  cases  of  ambiguity:  1st.  Where    Two  cases 
the  middle  term  is  equivocal ;  that  is,  when  used     istcase. 
in   different  senses   in  the  two  premises.      For 
example  : 


"  Light  is  contrary  to  darkness  ; 
Feathers  are  light ;  therefore, 
Feathers  are  contrary  to  darkness." 


Example. 


2d.  Where  the  middle  term  is  not  distrib-  2d  case. 
uted  ;  for  as  it  is  then  used  to  stand  for  a  part 
only  of  its  signijicates,  it  may  happen  that  one 
of  the  extremes  is  compared  with  one  part  of 
the  whole  term,  and  the  other  with  another  part 
of  it.     For  example  : 


Lgain 


Ebcample 


"  White  is  a  color ; 
Blark  is  a  color  ;  therefore, 
Black  is  white." 

"  Some  animals  are  beasts  ; 
Some  animals  are  birds  ;  therefore, 
Some  birds  are  beasts." 

The  middle 

3d.    The  middle  term,  therefore,  must  he  dis-  term  must  be 

once  distrib- 

trihuted,  once,  at  least,  in  the  premises  ;  that  is,       uted: 


90 


LOGIC. 


[book  I. 


and  cnce  is 
Bufficiuut. 


No  term  must 
be  dislri bil- 
led in  the 
conclusion 
which  was 
Dot  distribu- 
ted in  a 
premise. 


Examp'.c. 


Negalivft 

premises 

prove  noth- 


Esaniple. 


by  being  the  subject  of  a  universal,*  or  predi- 
cate of  a  negative  ;t  and  once  is  sufficient ; 
since  if  one  extreme  has  been  conapared  with  a 
part  of  the  middle  term,  and  another  to  the 
whole  of  it,  they  must  have  been  compared  with 
the  same. 

4th.  No  term  must  he  distributed  in  the  con- 
clusion which  was  not  distributed  in  one  of  the 
premises;  for,  that  would  be  to  employ  the 
whole  of  a  tei'm  in  the  conclusion,  when  you 
had  employed  only  a  part  of  it  in  the  premise  ; 
thus,  in  reality,  to  introduce  a  fourth  term. 
This  is  called  an  illicit  process  either  of  the 
major  or  minor  term.  J     For  example  : 

"  All  quadrupeds  are  animals, 
A  bird  is  not  a  quadruped  ;  therefore, 
It  is  not  an  animal."     Illicit  process  of  the  major. 

5th.  From  negative  premises  you  can  infer 
nothing.  For,  in  them  the  Middle  is  pronounced 
to  disagree  with  both  extremes ;  therefore  they 
cannot  be  compared  together :  for,  the  extremes 
can  only  be  compared  when  the  middle  agrees 
with  both ;  or,  agrees  with  one,  and  disagrees 
with  the  other.     For  example  : 

"  A  fish  is  not  a  quadruped  ;" 

"  A  bird  is  not  a  quadruped,"  proves  nothing'. 


*  Section  62.        f  Section  63.         X  Section  40, 


III.]  ANALYTICAL     OUTLINE.  91 


6lh.    If  one  premise  ba  negative,   the  conclu-  ifoneprem- 

,  .  f,         .  ,  .  ,  ise  is  nega- 

5/071  must  be  negative;  tor,  in  that  premise  the     tjve, the 
middle  term  is  pronounced  to  disagree  with  one    '=""'^'''*'"" 

^  c)  •^Yill  be  iiegar 

of  the  extremes,  and  in  the  other  premise  (which        ''^'®; 

of  course  is  affirmative  by  the  preceding  rule), 

to  agree  with  thi  other  extreme ;  therefore,  the 

extremes  disagreeing  with  each  other,  the  con- 

elusion  is  negative.     In  the  same  manner  it  may   andrecipro 

be  shown,  that  to  prove  a  negative  conclusion, 

one  of  the  premises  must  be  a  negative. 

By  these  six  rules   all   Syllogisms   are  to  be     what  fol- 
lows from 
tried;    and  from  them  it  will  be    evident,    1st,     these  six 

that  nothing  can  be  proved  from  two  particular 

premises ;  (since  you   will  then  have   either  the 

middle  term  undistributed,  or  an   illicit  process. 

For  example  : 

"  Some  animals  are  sagacious  ; 
Some  b?asts  are  not  sagacious  ; 
Some  beasts  are  not  animals.") 

And,  for  the  same  reason,  2dly,  that  if  one  of  ea inferenca 
the  premises  be  particular,  the  conclusion  must 
be  particular.     For  example  : 


"  All  who  fight  bravely  deserve  reward  ; 

"  Some  soldiers  fight  bravely  ;"  you  can  only  infer  that 

"  Some  soldiers  deserve  reward  :" 

for  to  infer  a   universal    conclusion   would    be 
an  illicit  process  of  the  minor.     But  from  two 


Example. 


92  LOGIC.  [b( 


rwouniver-  uiiivei'sal   Premises  you   cannot  always  infer  a 

sal  premises  .  i    /-i  i       •  t^  i 

doaotaiways  universal  Conclusion.     For  example: 

give  a  uni- 
versal con-  "  All  gold  is  precious  ; 

<=i"^i°'i'  All  gold  is  a  mineral ;  therefore, 

Some  mineral  is  precious.' 
I 
And  even  when  we  can  infer  a  universal,  we 

are  always  at  lihei'ty  to  infer  a  particular ;  since 

what  is  predicated  of  all  may  of  course  be  pre 

dicated  of  some. 


OF     FALLACIES. 

Definition  of      §  68.    By  a  fallacy  is  commonly  understood 

afaUacy.  ^  r  ■  i  •    i 

"  any  unsound  mode  oi  arguing,  which  appears 

to  demand  our  conviction,  and   to  be  decisive 

of  the  question  in  hand,  when  in  fairness  it  is 

Detection  of,  not."     In  the  practical  detection  of  each  indi- 

acuteness.    vidual   fallacy,   much    must   depend  on    natural 

and  acquired  acuteness ;  nor  can   any  rules  be 

given,  the  mere  learning  of  which  will  enable 

us  to  apply  them  with  mechanical  certainty  and 

Hints  and    rcadiuess  ;  but  still  we  may  give  some  hints  that 

rules  useful,  ^^.^j  j^^^  ^^  coiTcct  general  views  of  the  subject, 

and  tend  to  engender  such  a .  habit  of  mind,  as 

will  lead  to  critical  examinations. 
Same  of  Lo-       Indeed,  the  case  is  the  same  with  respect  to 
gicingenerai.  j^^^^^  j^^  general;   scarcely  any  one  would,  in 

ordinary   practice,    state    to    himself   either   his 


CHAP.   III.]  ANALYTICAL     OUTLINE.  93 


own  or  another's  reasoning,  in  syllogisms  at  full   Logic  tends 

,  ,  p.,..  -ii-i  ••!  to  cultivate 

length  ;  yet  a  lamiharity  with  logical  principles     ,j^(,its  of 

tends  very  much  (as  all  feel,  who  are  really  well  '^i'^™*""' 

acquainted  with  them)  to  beget  a  habit  of  clear 

and  sound  reasoning.     The   truth  is,  in  this  as 

in  manv  other  thinsrs,  there  are  processes  sfoinsr    Thehabu 

fixed,  we 

on   in  the  mind   (when  we    are  practising  any  naturally  foi- 
thing  quite   familiar  to   us),  with   such  rapidity    processes. 
as  to  leave  no  trace   in  the  memory ;  and  we 
often  apply  principles  which  did   not,  as  far  as 
we  are  conscious,  even  occur  to  us  at  the  time. 


§  69.   Let  it  be    remembered,   that   in   every   conclusion 

f,  .  1       •       11  II  follows  from 

process  oi   reasoning,  logically  stated,  the  con-   t^v^antece- 

clusion  is  inferred  from  two  antecedent  propo-    <ient  prem- 
ises. 
sitions,  called  the  Premises.     Hence,  it  is  man- 
ifest, that  in  every  argument,  the  fault,  if  there    Faiiac>-,  if 

,  ...  any,  either  in 

be  any,  must  be  either,  the  premises 

1st.  In  the  premises  ;  or, 
2d.  In  the  conclusion  (when  it  does  not  follow    orconciu- 

-  ,  ,  sion,  or  both, 

irorn  them) ;  or, 

3d.  In  both. 

In  every  fallacy,  the  conclusion  either  does  or 
does  not  follow  from  the  premises. 

When  the   fault   is   in  the  premises  ;  that  is,  when  in  the 
when  they  are  such  as  ought  not  to  have  been     p*"®™'^®^' 
assumed,  and  the  conclusion  legitimately  follows 
from  them,  the  fallacy  "s  called  a  Material  Fal- 


94  Logic.  [book  i. 

lacy,  because  it  lies  in  the  matter  of  the  argu- 
ment. 
When  in  the      Where  the  conclusion  does  not  follow  from 

conclusion.     ^,  ...  .^  ,  ,         „ 

the  premises,  it  is  manifest  that  the  fault  is  in 
the  reasoning,  and  in  that  alone:  these,  there- 
fore, are  called  Logical  Fallacies,  as  being  prop- 
erly violations  of  those  rules  of  reasoning  which 
it  is  the  province  of  logic  to  lay  down. 
When  in  When  the  fault  lies  in  both  the  premises  and 
reasoning,  the  fallacy  is  both  Material  and  Logical 


both. 


Rules  for         §  70.  In  examining  a  train  of  argumentation, 

examining  a^  ,    •        -c  r  i\  i  •  •. 

train  of  iu--    ^^  ascertain  it  a  iallacy  have  crept  into  it,  the 

guraent.  foUowing  poiuts  would  naturally  suggest  them- 
selves : 

tstRuie.  1st.  What  is  the  proposition  to  be  proved? 
On  what  facts  or  truths,  as  premises,  is  the  ar- 
gument to  rest  ?  and,  What  are  the  marks  of 
truth  by  which  the  conclusion  may  be  known  ? 

SdUuie.  2d.  Are  the  premises  both  true?  If  facts,  are 
they  substantiated  by  sufficient  proofs  ?  If  truths, 
were  they  logically  inferred,  and  from  correct 
premises? 

3d  Rule.  3d.  Is  the  middle  term  what  it  should  be,  and 

the  conclusion  logically  inferred  from  the  prem- 
ises ? 
Suggestions       These  general  suggestions  may  serve  as  guides 

serve  as       .  .    . 

guides,      ^^  examining  arguments  lor  the  purpose  of  de- 


CHAP,   in.]  ANALYTICAL     OUTLINE.  95 

tecting   fallacies ;    but   however  perfect  general     to  detect 
rules  may  be,  it  is  quite  certain  that  error,  in 
its  thousand  forms,  will  not  always  be  separated 
from  truth,  even  by  those  who  most  thoroughly 
understand  and  carefully  apply  such  rules 

CONCLUDING     REMARKS. 

§  71.  The  imperfect  and  irregular  sketch  which       Logic 

corresponds 

has  here  been  attempted  of  deductive  logic,  may     ^nh  the 
suffice  to  point  out  the  general  drift  and  purpose  ''eason"ib's  m 

^  "^  '^        '^  Geometry. 

of  the  science,  and  to  show  its  entire  correspond- 
ence with  the  reasonings  m   Geometry.      The 
analytical  form,  which    has  here  been  adopted.    Analytical 
is,  generally  speaking,  better  suited  for  introdu-       "™' 
cing  any  science  in  the  plainest  and  most  inter- 
esting form ;  though  the  synthetical  is  the  more    synthetical 
regular,  and  the  more  compendious  form  for  sto- 
ring it  up  in  the  memory. 


§  72.  It  has  been  a  matter  about  which  wri-    induction: 

1-1  T£r         1         1      .1  I    •  does  it  form 

ters  on  logic  have  dinered,  whether,  and  in  con-     a  part  of 
formity  to   what   principles.  Induction    forms   a       °^^ 
part  of  the  science ;  Archbishop  Whately  main-    whateiy's 
taming  that  logic  is  only  concerned  m  inierrmg 
truths  from  known  and  admitted  premises,  and 
that  all  reasoninff,  whether  Inductive  or  Deduc- 
live,  is  shown  by  analysis  to  have  the  syllogism 


96  LOGIC.  [book  I. 

Mill's  views,  for  its  type  ;  while  Mr.  Mill,  a  writer  of  perhaps 
greater  authority,  holds  that  deductive  logic  is 
but  the  carrying  out  of  what  induction  begins ; 
that  all  reasoning  is  founded  on  principles  of  in- 
ference ulterior  to  the  syllogism,  an.d  that  the 
syllogism  is  the  test  of  deduction  only. 

Without    presuming    at  all    to    decide  defini- 
tively a  question  which  has  been  considered  and 

Reasons  for  passcd  upou  by  two  of  the  most  acute  minds  of 

the  course 

taken.  the  age,  it  may  perhaps  not  be  out  of  place  to 
state  the  reasons  which  induced  me  to  adopt 
the  opinions  of  Mr.  Mill  in  view  of  the  par- 
tfcular  use  which  I  wished  to  make  of  logic. 

reading  Ob-       §  73.    It  was,  as    stated  in  the  general  plan, 

jects  of  the 

plan-       one  of  my  leading  objects  to  point  out  the  cor- 
respondence between  the  science   of  logic  and 
the   science  of  mathematics  :   to  show,   in  fact. 
To  show  that  that  mathematical  reasoning  conforms,  in  every 

mathemati-  ,  .  ,  ^  ,        .  ... 

cai  reasoning  I'sspcct,  to  the  sti'ictcst  rulcs  01  logic,  and  IS  in- 
conformsto  deed  but  logic  applied  to  the  abstract  quantities, 

Jo^ical  rules,  o  i  j. 

Number  and  Space.     In  treating  of  space,  about 

which  the  science  of  Geometry  is  conversant,  we 

shall  see  that  the  reasoning  rests  mainly  on  the 

Axioms,  how  axiouis,  and  that  these  are  established  by  induc- 

eetablished.     ,•  mu  r  •  i  •    u 

tive  processes.  1  he  processes  oi  reasoning  which 
relate  to  numbers,  whether  the  numbers  are  rep- 
resented by  figures  or  letters,  consist  of  two  parts. 


CHAP.  III.]  ANALYTICAL     OUTLINE.  97 

1st.  To  obtain  formulas  for,  that  is,  to  express 
in  the  language  of  science,  the  relations  between 
tlie  quantities,  facts,  truths  or  principles,  what-  Two  pans  of 

the  reasoning 

ever  they  may  be,  that  form  the  subject  of  the     process. 
reasoning ;  and, 

2dly.  To  deduce  from  these,  by  processes 
purely  logical,  all  the  truths  which  are  implied 
in  them,  as  premises. 


§  74.    Before  dismissing  the    subject,   it  may    ah  inauo- 

tion  may  be 

be  well  to  remark,  that  every  induction   may  thrown  into 
be  thrown  into  the  form  of  a  syllogism,  by  sup-       JJ-^^g 
plying  the  maior  premise.     If  this  be  done,  we  ^yi'ogism.by 

t'  J       ^  •>  t-  admitting  a 

shall  see  that  something  equivalent  to  the  uni-  proper  major 

r  •  r      1  f  Ml  premise. 

jormity  oj  the  course  of  nature  will  appear  as 
the  ultimate  major  premise  of  all  inductions  ; 
and  will,  therefore,  stand  to  all  inductions  in 
the  relation  in  which,  as  has  been  shown,  the 
major  premise  of  a  syllogism  always  stands  to 
the  conclusion ;  not  contributing  at  all  to  prove 
it,  but  being  a  necessary  condition  of  its  being 
proved.  This  fact  sustains  the  view  taken  by 
Mr.  Mill,  as  stated  above;  for,  this  ultimate  ma-     now  this 

.         .  ^        .       .  .     f.         major  prera 

jor  premise,  or  any  substitution  lor  it,  is  an  inter-   iseia  obtain- 
ence  by  Induction,  but  cannot  be  arrived  at  by        ^^ 
means  of  a  syllogism. 

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BOOK    II. 
MATHEMATICAL  SCIENCE. 


CHAPTER    I. 

QUANTITY  ANB  MATHEMATICAt,  SCIENCE  DEFINED— DIFFERENT  KINDS  OP  QUAN- 
TITY— LANGUAGE  OF  MATHKMATICS  EXPLAINED — SUBJECTS  CLASSIFLED— UNIT 
OF  MEASURE  DEFINED — MATHEMATICS  A  DEDUCTIVE   SCIENCE. 


QUANTITY. 

§  75.    Quantity  is  any  tiling  which   can   be    Quantity 
increased,  diminislied,  and  measnred. 


§  76.   Mathematics   is  the   science  of  quan-  Mathematics 
tity;    that   is,  the   science   Avhich   treats   of  the 
measures  of  quantities  and  of  their  relations  to 
each  other. 


§  77.   There  are  two  kinds  of  quantity;  Num-    Kinds  of 

1  T    o  quantity. 

ber  and  bpace. 


N"  U  51  B  E  K . 
§  78.   A    NUMBER   is  a  unit,  or  a  collection     Number 

defined. 

of  units. 


100  MATHEMATICAL     SCIENCE.  [BOOK    II. 

Abstract.  AlST   ABSTRACT    NUMBER    is    One    whose    Unit    is 

not  named;  as,  one,  two,  three,  &c. 

Denominate.        A    DENOMIITATE   NUMBER   is   One   AvllOSe   Unit   is 

named;   as  three  feet,  three  yards,  thi'ee  pounds. 

Such  numbers  are  also  called  concrete  numbers. 

How  we  Ob-      How  do  Tvc  acquire  our  first  notions  of  num- 

of  number,  bers  ?     Bj  first  presenting  to  the  mind,  through 

the   eye,   a   single    thing,   and    calling    it    one. 

Then  presenting  two  things,  and  naming  them 

two  ;    then    three    tilings,    and    naming    them 

three;   and  so  on   for  other  numbers.      Thus, 

It  is  done  by  ^6   acquirc  primarily,    in  a   concrete    form,   our 

perception    g]gjjjgjj|-g^j.y   ^otious   of   number,   by   perception, 

and        comparison,  and   reflection ;    for,  we    must  first 

reflection.  -^  '  '  ^ 

perceive  how  many  things  are  numbered;  then 
compare  what  is  designated  by  the  word  one. 
Reasons,  with  what  is  designated  by  the  words  two, 
three,  &o.,  and  then  reflect  on  the  results  of 
such  comparisons  until  we  clearly  apprehend 
the  difference  in  the  signification  of  the  words. 
Haying  thus  acquired,  in  a  concrete  form,  our 
conceptions  of  numbers,  we  can  consider  num- 
bers as  separated  from  any  particular  objects. 
Two  axioms  ^^^^  i\\\\^  form  a  conception  of  them  in  the  ab- 

necessary  for 

the  forma-   stract.      We    require   but    two   axioms    for   the 

tion  of  num- 
bers,      formation  of  all  numbers : 

1st  axiom.       Ist.  That  oue  may  be  added  to  any  number, 

and    that    the    number   which    results   will    be 


CHAP.   I.]  DEFIlSriTIOivTS.  101 

greater  by  one  than  the  number  to  which  the 
one  was  added. 

2d.  That  one  may  be  divided  into  any  num-   2d  axiom, 
ber  of  equal  parts.  ^ 

§  79.   Under  Xumber,  we  have  four  species,  or  Four  kinds 

of  number. 

subdivisions,  each  differing  from  the  other  three, 
in  the  unit  of  its  base :  thus, 

1.  Abstract  Number,  wlien  the  base  is  the  ab-    Abstract, 
stract  unit  one : 

3.  Number  of  Currency,  Avhen  the   base  is  a    Currency, 
unit  of  currency,  as  one  dollar: 

3.  Number  of  Weight,  when  the  base  is  a  unit     weight 
of  weight,  as  one  pound : 

4.  Number  of  Time,  when  the  base  is  a  unit      Time. 
of  time,  as  one  day. 

Hence,   in    number,   we   have    four   kinds   of  Four  kinds 

of  units, 

units:  Abstract  Units ;  Units  of  Currency;  Units 
of  Weight :  and  Units  of  Time. 


SPACE. 

§  80.   Space   is  indefinite  extension.     We   ac-      space 

defined. 

quire  our  ideas  of  it  by  obserAang  that  parts 
of  it  are  occupied  by  matter  or  bodies.  This 
enables  us  to  attach  a  definite  idea  to  the  word 
place.  We  are  then  able  to  say,  intelligibly, 
that  a  point  is  that  which   has  place,  or  posi-     a  point. 


102 


MATHEMATICAL     SCIEIirCE.  [BOOK   II. 


tion    in   space,  Avithout   occupying  any   part   of 

it.     Having  conceived  a  second  point  in  space, 

we   can   nnderstand   the   important   axiom,   "A 

Axiom  con-  Straight    line   is   tlie   shortest   distance   between 

Btrai^'^hfiiue  ^^^  points*"   and  this  line  we  call  length,  or  a 


dimension  of  space. 


§  81.  If  we  conceive  a  second  straight  line 
to  be  drawn,  meeting  the  first,  but  lying  in  a 
direction  directly  from  it,  we  shall  liave  a  sec- 
defliied.  ond  dimension  of  space,  which  we  call  hreadtli. 
If  these  lines  be  prolonged,  in  both  directions, 
they  will  include  four  portions  of  space,  winch 
make  up  what  is  called  a  plane  surface,  or  plane : 
hence,  a  plane  has  two  dimensions,  length  and 
breadth.  If  now  we  draAv  a  line  on  either  side 
of  this  plane,  we  shall  have  another  dimen- 
sion of  space,  called  thiclcness :  hence,  space  has 
three   dimensions — length,   breadth,   and    thick- 


A  plane 
deflued. 


Space  has 
three  dimen- 
sions. 


Figure 
defined. 


Line  de- 
fined. 


§  83.   A  portion  of  space  limited  by  bounda- 
ries,  is   called   a   Figure.      If    such   portion   of 
space  have  but  one  dimension,  it  is  called  a  line, 
and  may  be  limited  by  two  points,  one  at  each 
Two  kinds  extremity.     There  are  two  kinds  of  lines,  straight 
of  lines:     ^^^^-^  curved.     A  straight  line,  is  one  which  does 

straight  and  ° 

curved.     j^q|;  change  its  direction  between  any  two  of  its 


Difference. 


CHAP.  I.]  SPACE.  103 

points,  and  a  curved  line  constantly  changes  its 
direction  at  every  point. 

§  83.   A  portion  of  space  having  two  dimen-    surface : " 
sions  is  called  a  surface.     There  are  two  kinds 
of    surfaces — Plane   Surfaces   and   Curved    Sur-      cumTd. 
faces.     With  the  former,  a  straight  line,  having 
two    points   in   common,   will    always    coincide, 
however  it  may  be  placed,  while  with  the  latter 

''  -^  Boundaries 

it  will    not.      The    boundaries  of   surfaces  are  of » ^"'f^ce. 
lines,  straight  or  curved. 

§  84.   A  limited  portion  of  space,  having  three    volumes, 
dimensions,  is    called   a    Volume.      All   volumes  Boundaries, 
are  bounded  by  surfaces,  either  plane  or  curved. 

§  85.   An"  angle  is  the  amount  of  divergence 
of  two  lines,  of  two  planes,  or  of  several  planes, 

'  ^  '  ^  '       Angles. 

meeting  at  a  point;  and  is  measured,  like  other 
magnitudes,  by  comparing  it  with   its   unit  of 
measure.     Hence,  in  space,  we  have  four  units,    measure, 
differing  in  kind : 

1.  Linear    Units,    for     the     measurement    of      Linear, 
lines ; 

2.  Units  of  Surface,  for  the  measurement  of     surface, 
surfaces ; 

3.  Units  of  Volume,  for  the  measurement  of     volume, 
volumes ;  and 


104  MATHEMATICAL     SCIEKCE.  [BOOK    II. 

Angle.  4.  Units  of  Angles,  for  the  measurement  of 

angles. 

Eight  units.  §  86.  Besides  the  eight  kinds  of  units,  four 
of  number  and  four  of  space,  embraced  in  the 
above  classification,  and  in  which  the  units  of 
each  class  are  connected  by  known  laws,  there 
are  yet  isolated  denominate  numbers,  such  as 
five  chairs,  six  horses,  seven  things,  &c.,  which 

Unit?  with-  do  uot  admit  of  classification,  because  they  have 

out  law. 

no  law  of  formation.  Neither  does  this  classi- 
units.  fication  include  the  Infinitesimal  Units,  which 
are  specially  treated  of  in  Chapter  V.,  Book  II., 
and  which  are  the  elements  of  a  very  important 
branch  of  Mathematical  Science. 


Language        §  87.   The  language  of  Mathematics  is  mixed. 

mathema-  Althougli  composcd  mainly  of  symbols,  which 
are  defined  Avith  reference  to  the  uses  which 
are  made  of  them,  and  therefore  have  a  pre- 
cise signification;  it  is  also  composed,  in  part, 
of    words    transferred    from    our   common    lan- 

SjTnbois     miaare.     The  symbols,  although  arbitrary  signs, 

general.       &       ^  J  '  o  J        b      > 


are,  nevertheless,  entirely  general,  as  signs  and 

instruments  of  thought;  and  when  the  sense  in 

Avhich  they  are  used  is  once  fixed,  by  definition. 

Sense  un-   they  preserve    throughout    the    entire    analysis 

changed. 

precisely  the  same  signification.      The  meaning 


CHAP.    I.]       LANGUAGE     OF    MATHEMATICS.  105 


of  the  words  borrowed  from  our  common  vocab-  Words  bor- 
ulary  is  often  modified,  and  sometimes  entirely    common 
changed,  when  the  words  are  transferred  to  the  are  mocimed 
lano-uao-e  of  science.     They  are  then  used  in  .^ ''"f' ^^ed  in  a 

°       ^  ''  technical 

particular  sense,  and  are  said  to  have  a  technical      ^®'^^°- 
signification. 


§  88.  It  is  of  the  first  importance  that  the  Lan-uage 
elements  of  the  language  be  clearly  understood,  untostood : 
— that  the  signification  of  every  word  or  sym- 
bol be  distinctly  a^iprehended,  and  that  the  con- 
nection between  the  thought  and  the  word  or 
symbol  which  expresses  it,  be  so  well  estab- 
lished  that   the   one   shall   immediately  suffg-est  ,, 

•^         °°  Matliemati- 

the  other.     It  is  not  possible  to  pursue  the  sub-  *="'  reason- 
ings require 
tie  reasonings  of  Mathematics,  and  to  carry  out        it- 

the  trains  of  thought  to  which  they  give  rise, 
without  entire  familiarity  with  those  means 
which  the  mind  employs  to  aid  its  investiga- 
tions.    The  child  cannot  read  till  he  has  learned  ^ 

Cannot  nse 

the  alphabet ;  nor  can  the  scholar  feel  the  deli-     ^"^  ^'*°' 

'-  guage  well 

cate  beauties  of  Shakspeare,  or  be  moved  by  the      *'''  ^'^ 

^  -^  know  it. 

sublimity  of  Milton,  before  studying  and  learn- 
ing the  language  in  which  their  immortal 
thoughts  are  clothed. 


§  89.   All  Quantities,  whether  abstract  or  con-   Quantities 
Crete,   are,   in    mathematical    science,   presented    ^'*^ ''"P"'®" 

5* 


106  MATHEMATICAL     SCIElSrCE.  [iJOOK   II. 

sentedby    to   the   miud   bj  arbitrary   symbols.     Thev   are 

symbols ;         _  ^ 

and  are  opei--  vieAved  and  Operated  on  through  these  symbols 

ated  on  bj'^  . 

these  gym-   which   represent   them;    and   all   operations   are 

indicated    by   another    class   of    symbols    called 

Signs.      signs.       These,     combined     with     the     symbols 

whatconsti- ^^^^^^^^    represent    the    quantities,    make    np,   as 

i!rf!„'  -f    ^^'6   hn\e   stated   above,   the   pure    mathematical 

lan<juage.  '  i 

language ;    and    this,   in   connection   with    that 
which  is  borrowed  from  our  common  language, 
forms    the    language    of    matliematical    science. 
This    language    is   at   once   comprehensive    and 
Its  nature,   accurate.     It  is  capable  of  stating  the  most  gen- 
eral propositions,  and  of  presenting  to  the  mind, 
in   their  proper  order,  all  elementary  principles 
What  it  ac-  Connected   with    their   solution.      By   its   gener- 
cpmpiMes.  jj^jj^y  ^j.    j.gj^(3]5gg    q^qj.    the   whole    field    of    the 

pure  and  mixed  sciences,  and  gathers  into  con- 
densed forms  all  the  conditions  and  relations 
necessary  to  the  development  of  particular  facts 
and  universal  truths.  Thus,  the  skill  of  the 
analyst  deduces  from  the  same  equation  the  ve- 
Extentand  locity  of  an  apple  falling  to  the  ground,  and  the 

power  of 

Analysis,    verification  of  the  law  of  universal  gravitation. 

LANGUAGE   OF   MATHEMATICS. 

Lan^ma^e        §  ^^-   The  language  of  Mathematics  embraces, 
„  °^  1st.  Parts  of  our  written  and  spoken  language: 

Mathema-  '  o      o    ^ 

^^^^-  2d.  The  language  of  Figures , 


CHAP.   I.]       LANGUAGE    OF    MATHEMATICS.  107 

3d.  The  language  of  Lines — straight  and  curved ;      Lines. 

4th.  The  language  of  Letters;  and  these  forms     Letters, 
of  language  determine  the  classification  of  the 
branches  of  the  Science  of  Mathematics. 

LANGUAGE   OF   NUMBEE — ARITHMETIC. 

§  91.    The   ten   characters,   called  figures,   are    Language 

of 

the  alphabet  of  the  language  of  number.  The  Number. 
various  ways  in  which  they  are  combined,  form- 
ing the  exact  and  copious  language  of  compu- 
tation, Avill  be  fully  explained  under  the  head 
of  Arithmetic,  in  a  chapter  exclusively  devoted 
to  the  consideration  of  numbers,  their  laws  and 
their  language. 

LANGUAGE   OF    LINES — GEOMETRY. 

§  92.   Lines,  straight  and  curved,  are  the  ele-    Language 

of 
ments   of    the   pictorial    language   applicable   to   Geometry. 

space.     The   definitions  and   axioms   relating  to 

space,  and  all  the  reasonings  founded  on  them, 

make   up  the   science  of  Creometry,  Avhich   will 

be  ftilly  treated  under  its  proper  head. 

LANGUAGE   OF  LETTERS — ANALYSIS. 

§  93.   Analysis  is  a  general  term  embracing    Analysis. 
all  the  operations  which   can  be  performed  on 


108  MATHEMATICAL     SCIENCE.  [BOOK   II. 

quantities  when  represented  by  letters.     In  this 
branch  of  mathematics,  all   the  quantities  con- 
Quantities   sidered,  whether   abstract   or  concrete,  are   rep- 
represented  TIT,  nl  1T1J  TJ_1 

by  letters,  resented  by  letters  oi  the  alphabet,  and  the 
operations  to  be  performed  on  them  are  indi- 
cated   by   a    few   arbitrary   signs.      The    letters 

Symbols,  and  signs  are  called  Symbols,  and  by  their 
combination  w^e  form  the  Algebraic  Notation 
and  Language. 

Analysis        §  ^^^  Analysis,    in    its    simplest    form,  takes 
Algebra;    the  name  of  Algebra.     Analytical  Geometry,  the 

Analytical  "  ''  •" 

Geometry.   Differential   and   Integral  Calculus,  extended  to 
Calculus,    include  the  Theory  of  Variations,  are  its  higher 
and  most  advanced  branches. 

TermAnaiy-      §  95.   The  term  Analysis  has  also  another  sig- 
nification.    It  denotes  the  process  of  separating 
Its  nature,   any  complcx  whole  into  the  elements  of  which 
Synthesis    it  is  coiiiposed.     It   is   opposed   to  Synthesis,   a 

defined. 

term  which  denotes  the  processes  of  first  con- 
sidering the  elements  separately,  then  combin- 
ing them,  and  ascertaining  the  results  of  the 
combination. 

Analytical  The  Analytical  method  is  best  adapted  to  in- 
method.     ye^tigation,  and  the  presentation  of  subjects  in 

Synthetical  their  general  outlines;  the  Synthetical  method 
is  best  adapted  to  instruction,  because  it  exhib- 


CHAP.   I.]  II^TFIITITESIMAL    CALCITLTJS.  109 

its  all  the  parts  of  a  subject  separately,  and  in    Analysis, 
their    proper   order   and    connection.      Analysis 
deduces  all  the   parts  from  a  whole:    Synthesis    Synthesis. 
forms  a  whole  from  the  sejjarate  parts. 

§  96.   Arithmetic,  Algebra,  and   Geometry  are  Arithmetic, 
the   Elementary  branches  of  Mathematical   Sci-    ,,  "^  l^' 

J  Geometry, 

ence.       Every    truth    which    is    established    by  elementary 

•^  -^      branches. 

mathematical  reasouiug,  is  developed  by  an 
arithmetical,  geometrical,  or  analytical  process, 
or  by  a  combination  of  them.  The  reasoning 
in  each  branch  is  conducted  on  principles  iden- 
tically  the    same.      Every   sign,    or    symbol,   or   Reasoning 

.,.,  ,.  -t       ->    n       ^  ^^^  same. 

technical  Avord,  is  accurately  denned,  so  that  to 
each  there  is  attached  a  definite  and  precise 
idea.     Thus,  the  language  is  made  so  exact  and    Language 

exact. 

certain,  as  to  admit  of  no  ambiguity. 

II^FIlSriTESIMAL     CALCULUS. 

8  97.   The  language  of  the  Infinitesimal  Cal-    Language 

^  ®      °  (ifihe 

cuius  is  very  simple.     Its  chief  element   is   the  inflniteshnai 

Calculus. 

letter  d,  which,  when  written  before  a  quantity, 
denotes  that  that  quantity  increases  or  decreases 
according  to  the  law  of  continuity,  and  the  ex- 
pression thus  arising  is  one  link  in  that  law. 
Thns,  dx  denotes  that  the  quantity  represented  What  does 

dx  denote. 

by  X,  changes  according  to  the  law  of  continuity, 
and  that  dx  is  the  unit  of  that  change. 


110  MATHEMATICAL     SCIENCE.  [BOOK   II. 


PURE     MATHEMATICS. 

Pure  Mathe-      §  98.   The  Pure  Mathematics  embraces  all  the 

matics. 

principles  of  the  science,  and   all   the  processes 

by   which   those   principles   are   developed   from 

Number  and  the  abstract  quantities,  Number  and  Space.     All 

Space. 

the  definitions  and  axioms,  and  all  the  truths 
deduced  from  them,  are  traceable  to  these  two 
sources. 


Mathema-        §  99.   Mathematics,   in    its   primary   significa- 

tics,  as  used 

by  the  an-  tiou,  as  uscd  by  the  ancients,  embraced  every 
acquired  science,  and  was  equally  applicable  to 
all  branches  of  knowledge.  Subsequently  it 
was  restricted  to  those  brandies  only  which 
Avere  acquired  by  severe  study,  or  discipline,  and 

Embraced   its  votarics  Were   called   Disciples.      Those   sub- 

all  subjects  . 

which  wore  jects,  therefore,  which  required  patient  mvesti- 
in  their  iia-  ga^tiou,  cxact  reasoning,  and  the  aid  of  the  ma- 
*'^'^'  thematical  analysis,  were  called  Disciplinal  or 
Mathematical,  because  of  the  greater  evidence 
in  the  arguments,  the  infallible  certainty  of  the 
conclusions,  and  the  mental  training  and  de- 
velopment which  srxh  exercises  produced. 

Pure  Mathe-      §  100.   The    Purc    Mathematics   is   based    on 
definitions   and   intuitive  truths,  called   axioms, 

What  are  its 

foundations,  which  are  inferred  from  observation  and  expe- 


CHAP.   I.]  PURE     MATHEMATICS.  Ill 

rience ;  that  is,  observation  and  experience  fur-    Premises, 
nish    tl]e   information   necessary   to    such   intui- 
tive  inductions.*      From    these   definitions   and 
axioms,  as  premises,  all  the  trnths  of  the  science  Reasoning, 
are  established  by  processes  of  deductive  reason- 
ing;  and   there   is   not,  in   the  whole   range  of   its  tests  of 
mathematical  science   any  logical   test   of  truth, 
lut   in   a   conformity   of  the   conclusions   to   the  what  they 

are. 

definitions  and  axioms,  or  to  such  j^^'i^^dples  or 
propositions  as  have  been  established  from  them. 
Hence,  we  see,  that  the  science  of  Pure  Mathe-  in  what  the 

,.  I'l  -i.  !••/>•  1       science  con- 

matics,  Avhich   consists   merely  m   inferiing-,   by       gj^^g^ 

fixed  rules,  all  the  truths  Avhich  can  be  deduced 

from    given    premises,    is    purely    a    Deductive    i«  purely 

Deductive. 

Science.  The  precision  and  accuracy  of  the 
definitions ;  the  certainty  which  is  felt  in  the 
truth  of  the  axioms ;  the  obvious  and  fixed  re-  Precision  of 

its  language. 

lation  betAveen  the  sign  and  the  thing  signified ; 
and  the  certain  formulas  to  which  the  reason- 
ing processes  are  reduced,  have  given  to  mathe- 

Exact, 

niatics  the  name  of  '-Exact  Science."  '  science. 

§  101.  We  have  remarked  that  all  the  rea-  ah  rrason- 
sonings  of  mathematical  science,  and  all  the  definitions 
truths  which  they  establish,  are  based  on  the  """^  ''^^''"''• 
definitions  and  axioms,  Avhich  correspond  to  the 

*  Section  37. 


112  MATHEMATICAL     SCIEXCE.  [BOOK    II. 

major  premise  of  the  syllogism.     If  the  resem- 
blance which   the  minor  premise  asserts  to  the 
Relations    middle   term  were   obvions  to  the   senses,  as  it 
"'is   in   the   proposition,    "Soci-ates   was   a   man," 
or  were  at  once  ascertainable  by  direct  observa-' 
tion,  or  were  as  evident  as  the  intnitiye  trnth, 
"A  whole  is  eqnal  to  the  sum  of  all  its  parts;" 
Deductive   there  would   be  no   necessity  for  trains  of  rea- 
necessuiy.   soning,  and  Deductive  Science  wonld  not  exist. 
Trains  of    Trains  of  reasoning  are  necessary  only  for  the 
reasoiimg.  ^_^^^  ^^  extending  the  definitions  and  axioms  to 
What  they  otlicr   cases   in   which   we   not   only  cannot   di- 
accomp  is  .  j.gg^jy  observe  what  is  to  be  proved,  but  cannot 
directly   observe    even    the    mark   which    is    to 
prove  it. 

Syllogism,       §  103.   Although  the  syllogism  is  the  ultimate 
^^of'lieduc-'''^  test  in  all  deductive   reasoning   (and   indeed  in 
*^°°"       all  inductive,  if  Ave  admit  the  uniformity  of  the 
course  of  nature),  still  w^e  do   not  find   it   con- 
venient or  necessary,  in  mathematics,  to  throw 
every  proposition  into  the  form  of  a  syllogism. 
.  .  ,      The   definitions   and   axioms,  and   the   propo- 

Axioms  and  ^       ^ 

definiii.ms,  gitions  established   from  them,  are  our  tests  of 

tests  of 

truth.      truth;   and   whenever  any  new  proposition  can 

be    brought    to   conform    to   any   one   of    these 

Apropos!-  tests,  it  is  regarded  as  proved,  and  declared  to 

tion :  when 

proved,     be  true. 


CHAP.   I.]  MIXED    MATHEMATICS.  113 

§  103.   When     general     formulas     have     been     When  a 
framed,   determining    the    limits   within    which   maybere- 
the   deductions    may   be   drawn    (that    is,   what    ^proved.^ 
shall  be  the  tests  of  truth),  as  often  as  a  new 
case  can   be   at   once   seen  to  come  within  one 
of  the  formulas,  the  principle  applies  to  the  new 
case,  and  the  business  is  ended.     But  new  cases    Trains  of 

reasoning: 

are  continually  arising,  which  do  not  obviously  whyncces- 
come   within   any   formula   that   will   settle   the          '^' 
questions   we   want   solved   in    regard   to   them, 
and  it  is  necessary  to  reduce  them  to  such  for- 
mulas.    Tliis  gives  rise  to  the  existence  of  the    They  give 

1         ,  .    1  r'*e  t^o  the 

science   oi    mathematics,    requiring   the   highest    ecicnceof 
scientific  genius  in  those  who  contributed  to  its     ^  ^^^.^^  ' 
creation,  and  calling  for  a  most  continued  and 
vigorous  exertion  of  intellect,  in  order  to  appro- 
priate it,  when  created. 

MIXED     MATHEMATICS. 

§  104.  The  Mixed  Mathematics  embraces  the  Mixed 
applications  of  the  principles  and  laws  of  tlie  tics. 
Pure  Mathematics  to  all  investigations  in  which 
the  mathematical  language  is  employed  and  to 
the  solution  of  all  questions  of  a  practical  na- 
ture, whether  they  relate  to  abstract  or  concrete 
quantity. 

8 


114  MATHEMATICAL     SCIENCE.  [HOOK   II. 


QUAN'TITT     MEASURED. 


Quantity.        §105.   Quantity  has  been  defined,  "anything 
which  can  be  increased,  diminished,  and  nieas- 
increased    ^^i-ed."     The  terms  increased  or  diminished,  are 


and 

diminished,  easily  nnderstood,  implying  merely  the  property 
of    being    made   larger   or   smaller.      The   term 
measured  is  not  so  easily  comprehended,  because 
it  has  only  a  relative  meaning. 
Measured.       The  term  "measured,"  applied  to  a  quantity, 
implies   the  existence  of  some   known  quantity 
wiiat  it     of  the  same  kind,  which  is  regarded  as  a  stand- 
'^'^'''     ard.    With  this  standard,  the  quantity  to  be  meas- 
ured is  compared  with  respect  to  its  extent  or 
standard:    magnitude.     To  such  standai-d,  Avhatever  it  may 
is  called     be,   we    give    the    name   of    iinity,   or    unit  of 
'^""^'      measure;   and  the  number  of  times  Avhich  any 
quantity   contains   its   unit   of   measure,   is   the 
numerical  value  of  the  quantity  measured.     The 
Magnitude:  extent  or  magnitude  of  a  quantity  is,  therefore, 
tivc.       merely  relative,  and  hence,  we  can  form  no  idea 
of  it,  except  by  the  aid  of  comparison.     Space, 
Space:     for  example,  is  entirely  indefinite,  and  we  meas- 

Indeflnite. 

lire  parts  of  it  by  means  of  certain  standards, 
Measure-    Called  measures ;  and  after  any  measurement  is 

ment  ascer- 
tains rcia-   completed,  we  have  only  ascertained  the  relation 

or  proportion  which  exists   between  the   stand- 
ard we  adopted  and  the  thing  measured.    Hence, 


CHAP.    I.]  QUAKTITT    MEASURED.  115 


measurement  is,   after  all,  but  a  mere  process  a  process  of 

„  .  compai'isoiL 

01  comparison. 


§  106.  The  quantities,  Number  and  Space,  are  but  Number  and 
vague  and  indefinite  conceptions,  until  we  compare   known  by 
them  with  their  units  of  measure,  and  even  these  *^°'^i'^"^<''*- 
units  are  arrived  at  only  by  processes  of  comparison. 
Comparison  is  the  great  means  of  discovering  the  comparison 
relations  of  things  to  each  other,  as  well  in  general     method, 
logic,  as   in   the   science  of  mathematics,  which 
develops   the  processes  by  which  quantities  are 
compared,  and  the  results  of  such  comparisons. 


§  107.  Besides   the   classification   of   quantity    Quantity, 
into  Number  and  Space,  we  may,  if  we  please, 
divide  it  inta  Abstract  and  Concrete.     An  ab-    Abstract, 
stract  quantity  is  a  mere  number,  in  Avhich  the 
unit  is  not  named,  and  has  no  relation  to  mat- 
ter or  to  the  kind  of  things  numbered.     A  con-    concrete. 
Crete   quantity  is  a  definite   object   or  a  collec- 
tion of  such   objects.      Concrete   quantities   are 

expressed   by  numbers  and  letters,  and  also  by  iiow  repre- 
sented, 
lines,  straight  and  curved.     The  number  "  three"    ^ 

^  °  Example 

is  entirely  abstract,  expressing  an    idea   having      of  the 

abstract. 

no  connection  with   things;    while   the  number 
"three  pounds  of  tea,"  or  "three  apples,"   pre-    Example 
sents  to  the  mind  an  idea  of  concrete  objects. 


concrete. 


So,  a  portion  of  space,  bounded  by  a  surface,  all 


116  MATHEMATICAL     SCIENCE.  [BOOK   II. 

the  points  of  wliicli  are  equally  distant  from  a 
Sphere     Certain  point  within   called  the  centre,  is  but  a 

mental  conception  of  form ;  but  regarded  as  a 
defined,     portion  of  space,  gives  rise  to  the  additional  idea 

of  a  named  and  defined  thins:. 


COMPAEISON"    OF     QUAIfTITIES. 

Mathematics      §  108.   We   liaYe   Seen   that  the  pure    mathe- 
HithNum-  niatics   are   concerned  with   the   two   quantities, 
Space      Number  and  Space.     We  have  also  seen,  that  rea- 
Keasoning  soning  nccessarilj  involves   comparison :    hence, 
comparison,  mathematical    reasoning    must   consist  in   com- 
paring the  quantities  wliich  come  from  Number 
and  Space  with  each  other. 


Two  qiianti-      §  ^^^'   -^^J    ^^^    quantities,    compared    with 
ties  can  sns-  qq^q]^  other,  must  necessarilv  sustain  one  of  two 

tain  but  two  '  •' 

relations,  relations:  tbey  must  be  equal,  or  unequal.  What 
axioms  or  formulas  have  we  for  inferring  the 
one  or  the  other? 


AXIOMS   FOR   IlfFERRIXG    EQUALITY. 

1.  Quantities  which  contain  the  same  nnit  an 
Formnias    equal  number  of  times,  are  equal. 
^  '''5.,  2.  Things  which  being  applied  to  each  other 

Equality.  o  o      1 1 

coincide,  are  equal  in  all  their  parts. 


CHAP.   I.]        COMPAEISGN    OF   QUANTITIES.  117 

3.  Things  wliicli  are  equal  to  tlie  same  thing 
are  equal  to  one  another. 

4.  A  whole  is  equal  to  the  sum  of  all  its  parts 

5.  If  eqnals  be  added  to  equals,  the  sums  are 
equal. 

6.  If  equals  be  taken  from  equals,  the  remain- 
ders are  equal. 

AXI03IS   FOR   IXFEimiXG    I]SrEQUALITY. 

1.  A  whole  is  greater  than  any  of  its  parts. 

3.  If  eqnals  be   added  to  uneqnals,  the  sums    Formulas 

for 

are  unequal.  inequality. 

3.  If  eqnals  be  taken  from  uneqnals,  the  re- 
mainders are  unequal. 

§  110.   We  have  thns  completed  a  very  brief    what  fea- 
tures have 
and     general     analytical     view     of     Mathema-       been 

tical  Science.  We  have  named  and  defined 
the  subjects  of  which  it  treats — and  the  forms 
of  the  language  employed.  We  have  pointed 
out  the  character  of  the  definitions,  and  the  na- 
ture of  the  elementary  and  intnitive  proposi- 
tions on  which  the  science  rests ;  also,  the  kind 
of  reasoning  employed  in  its  creation,  and  its 
divisions  resulting  from  tlie  use  of  different 
symbols  and  differences  of  language.  We  shall 
now  proceed  to  treat  the  branches  separately. 


sketched. 


S  o 

C3 

o  ^ 

1— 1 

CO 

r-<    iW 

Sd 

1-^  o 

1— ( 

(=1 

^"o 

w 

B  o 

e^ 

-^  o 

<^ 

'UIAP.  IX.]       ARITHMETIC FIRST     NOTIONS.  119 


CHAPTER    II. 

ARITHMETIC SCIENCE    AND    ART    OF    NUMBERS. 


SECTION    I. 


INTEGKAL    UNITS 


FIRST     NOTIONS     OF     NUMBERS. 

§  111.   There  is  but  a  single  elementary  idea  But  one  eio 

,  .  r  1  •       •         1         •  1  r     y        mentary  idea 

in  the  science  oi  numbers:  it  is  the  idea  of  the  lu numbers. 
UNIT  ONE.     There  is  but  one  way  of  impressing     Howim- 
this  idea  on  the  mind.     It  is  by  presenting  to    ^thTmm^d! 
the  senses  a  single  object ;    as,  one  apple,  one 
peach,  one  pear,  &;c. 


.5  112.     There   are   three   signs  by   means    of  Threesigns 

.                                ,  for  express- 

which  the  idea  of  one  is  expressed  and  commu-      ingu. 
nicated.     They  are, 

1st.  The  word  one.  a  word. 

2d.   The  Roman  character  I.  Roman 

o  1      mi        n  character-, 

od.  The  figure  I. 

'=                                                 .  Figure. 


120  MATHEMATICAL     SCIENCE  [bOOK  II 


New  ideas         §  113.  If  oiie  be  added  to  one,  the  idea  thus 

which  iii'isG 

by  adding    arising  is  different  from  the  idea  of  one,  and  is 

°^^'       complex.     This  new  idea  has  also  three  signs ; 

viz.   TWO,   II.,   and  2.     If  one  be  again   added, 

that  is,  added  to  two,  the  new  idea  has  likewise 

three   signs ;    viz.  three,   III.,   and  3.      These 

Collections  collections,   and   similar   ones,   arc   called    num- 
are  num-  ^ 

bers.       bers.    Hence, 

A  NUMBER  is  a  unit  or  a  collection  of  units. 


IDEAS  OF  NUMBERS  GENERALIZED. 

Ideas  of  §  114.  If  wc  begin  with  the  idea  of  the  num- 

generaiized.  ^cr  onc,  and   then  add  it  to  one,  making  two  ; 

and  then  add  it  to  two,  making  three  ;  and  then 

to  thi'ee,  making  four ;  and  then  to  four,  making 

How  formed,  fivc,  and  SO  On ;  it  is  plain  that  we  shall  form  a 

series  of  numbers,  each  of  which  will  be  greater 

Hnity  the    bv  onc  than  that  which  precedes  it.     Now,  one 

or  unity,  is  the  basis  of  this  series  of  numbers. 

Three  ways 

of  expressing  and  each   number   may  be    expressed   in    three 

them. 

ways  : 
1st  way.  1st.  By  the  words  one,  two,  three,  &c.,  of  our 

common  language  ; 
2d  way.  2d.  By  the  Romaii  characters ;  and, 

3d  way.  3d.  By  figures. 


CHAP.  II.]  .ARITHMETIC UNITY.  12,1 


notions  are 
complex. 


§  115.  Since  all  numbers,  whether  integral  or  AUanmbers 

^  .  ,  P  111  come  from 

rractional,  must  come  irom,  and  hence  be  con-       ^ne: 

nected  with,  the  unit  one,  it  follows  that  there 

is  but  one  purely  elementary  idea  in  the  science 

of  numbers.     Hence,  the  idea  of  every  number,    Hence  but 

11  I  r  •  /         1      11  1  one  idea  thai 

regarded  as  made  up  oi  units    (and  all  numbers  is  purely  eie- 
except  one  must  be  so  regarded  when  we  ana-     '^™'^''-^- 
lyze  them),  is  necessarily  complex.     For,  since     another 
the  number  arises  from  the  addition  of  ones,  the 
apprehension  of  it  is  incomplete  until  we  under- 
stand how  those  additions  were  made  ;  and  there- 
fore,   a   full   idea  of  the   number  is   necessarily 
com.plex. 

§  116.  But  if  we  regard  a  number  as  an  en- 
tirety, that  is,  as  an  entire  or  whole  thing,  as  an 
entire  two,  or  three,  or  four,  without  pausing  to     when  a 

,  ,  .  f,        ,  .    ,     .      .  ,  .  number  may 

analyze  the  units  oi  which  it  is  made  up,  it  may  ^^,  regarded 
then  be  regarded  as  a  simple  or  incomplex  idea ;  asmcompiex. 
though,  as  we  have  seen,  such  idea  may  always 
be  traced  to  that  of  the  unit  one,  which  forms 
the  basis  of  the  number. 


UNITY     AND     A     UNIT     DEFINED. 

§  117.   When  we  name  a  number,  as  twenty  what  is  ne- 
feet,  two  things  are  necessary  to  its  clear  appre-  ^''^*^'y'°'*^'' 

^  "^  '^  apprehension 

hension.  ©I  a  number 


122 


MATHEMATICAL     SCIENCE. 


[book  II. 


First. 


1st.    A    distinct    apprehension    of    the    single 

thing  which  forms  the  base  of  the  number ;  and, 

Second.         2d.  A  distinct  apprehension  of  the  number  oj 

times  which  that  thing  is  taken. 

The  basis  of       The  single  thing,  which  forms  the  base  of  the 

the  number  i  •  ii      i  t^    •  ii     j 

u  UNITY,  numbei',  is  called  unity,  or  a  unit,  it  is  called 
When  it  is  Unity,  when  it  is  regarded  as  the  primary  base 
called  UNITY,  ^j-  ^j^g  number ;  that  is,  when  it  is  the  final  stand- 
ard to  which  all  the  numbers  that  come  from  it 
are  referred.  It  is  called  a  unit  when  it  is  re- 
garded as  one  of  the  collection  of  several  equal 
thinsrs  which  form  a  number.  Thus,  in  the  ex- 
ample,  one  foot,  regarded  as  a  standard  and  the 
base  of  the  number,  is  called  unity;  but,  con- 
sidered as  one  of  the  twenty  equal  feet  which 
make  up  the  number,  it  is  called  a  unit. 


and  when  a 

UNIT. 


Abstract 
unit. 


OF  SIMPLE  AND  DENOMINATE  NUMBERS. 

§  118.  A  simple  or  abstract  unit,  is  one,  with- 
out regard  to  the  kind  of  thing  to  which  the  term 
one  may  be  applied. 
Denominate       ^  denominate  or  concrete  unit,  is  one  thing 
""''•       named  or  denominated  ;  as,  one  apple,  one  peach, 
one  pear,  one  horse,  &c. 


Number  has 
no  reference 


§  119.    Number,   as   such,   has  no  reference 
to  the  particular  things  numbered.     But  to  dis- 


CHAP.  II.]  ARITHMETIC ALPHABET.  123 


tinguish  numbers  which  are  applied  to  particular  to  the  things 
units  from  those  which  are  purely  abst.iact,  we 
call    the    latter  Abstract   or   Simple  Numbers,     simple 
and  the  former  Concrete  or  Denominate  Num-  Denominate. 
bers.      Thus,   fifteen    is  an   abstract   or  simple 
number,  because    the    unit    is    one ;   and  fifteen    Examples. 
pounds  is   a    concrete   or   denominate  number, 
because  its  unit,  one  pound,  is  denominated  or 
named. 


ALPHABET WORDS GRAMMAR. 

§  120.   The  term  alphabet,  in  its  most  general    Alphabet 
sense,  denotes  a  set  of  characters  which  form 
the  elements  of  a  written  language. 

When   any   one   of  these  characters,   or  any     vvonia. 
combination  of  them,  is   used  as  the  sign  of  a 
distinct  notion  or  idea,  it  is  called  a  word  ;  and 
the  naming  of  the  characters  of  which  the  word 
is  composed,  is  called  its  spelling. 

Grammar,  as  a  science,  treats  of  the  estab-    Grammai 
lished  connection  and  relation  of  words,  as  the 
signs  of  ideas. 

ARITHMETICAL     ALPHABET. 

§  121.    The  arithmetical  alphabet  consists  of  Arithmetical 

Alphabet 

ten  characters,  called  figures.     1  hey  are. 

Naught,  One,     Two,  Throe,  Four,    Five,     Six,    Seven,   Eight,   Nine, 

012345678       9 


124  MATHEMATICAL     SCIENCE.  [bOOK  II. 


and  each  may  be  regarded  as  a  word,  since  it 
stands  for  a  distinct  idea. 


WORDS — SPELLING  AND  READING  IN  ADDITION. 

ono  cannot  §  122.  The  idea  of  one,  being  elementary,  the 
e  ape  e  .  ^.j^^j.^^j^gj.  j  ^yhich  represents  it,  is  also  element- 
ary, and  hence,  cannot  be  spelled  by  the  other 
characters  of  the  Arithmetical  Alphabet  (§  121). 
But  the  idea  which  is  expressed  by  2  comes  from 

Spelling  by  the  addition  of  1  and  1  :  hence,  the  word  repre- 

the 

nritbraeticai  sented  by  the  character  2,   may  be   spelled  by 
Characters,    j  ^^^^  j_     rpj^^^^  j   ^^^^  j  ^^,g  2,  is  the  arithmet- 
ical spelling  of  the  word  two. 

Three   is   spelled   thus :    1  and  2   are  3 ;    and 
also,  2  and  1  are  3. 
Eiiampies.        YouY  is  spelled,  1  and  3  are  4  ;  3  and  1  are  4 ; 
2  and  2  are  4. 

Five  is  spelled,  1  and  4  are  5  ;  4  and  1  are  5 ; 
2  and  3  are  5  ;  3  and  2  are  5. 

Six  is  spelled,  1  and  5  are  6  ;  5  and  1  are  6 ; 
2  and  4  are  6 ;  4  and  2  are  6 ;  3  and  3  are  6. 

All  numbers  §  123.  In  a  similar  manner,  any  number  in 
"^ledina   ai'lthmctic  may  be  spelled;    and  hence  we  see 

Bimiiarway.  ^j-,^t  ^j-jg  process  of  spelling  in  addition  consists 
simply,  in  naming  any  two  elements  which  will 
make  up  the  number.     All  the  numbers  in  ad- 


CHAP     11. J  ARITHMETIC READINGS.  125 


ditiun  are  therefore  spelled  with  two  syllables. 

The  reading  consists  in  naming  only  the  word  Reading:  in 

which  expresses  the  final  idea.     Thus,  gists. 

0123456789  Examplea. 

1111111111 

One      two      three     four       five        six      seven    eiglit     nine      ten. 

We  may  now  read  the  words  which  express. 
the  first  hundred  combinations. 


READINGS. 

Read. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Two,  three, 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

foui-,  &c. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Three,  four, 

2 

2 

2 

2 

■   2 

2 

2 

2 

2 

2 

&c. 

123456789       10  Four,  five, 

3333333333 


&c. 


12345678       910       Five,six,&Q 
4444444444 

123456789       10  six,  seven, 

55555555       5       5 


&c. 


123456789       10  seven,  eight, 

6       6       6       6 6       6       6       6       6       6  *'''• 

123456789       10  Eight,nine, 

7777777777     ^''• 

123456789       10  Ninc,ten,&<% 

8888888888 


126 


MATHEMATICAL     SCIENCE. 


[book  II 


Ten,  eleyen,  1 

&c.  r» 


Eleven, 
twelve,  &c. 


8  9     10 

9  9       9 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

Example  for       §  124.  In  this  example,  beginning 

reading  in  •    i       i 

Addition,  at  the  right  hand,  we  say,  8,  17,  18, 
26 :  setting  down  the  6  and  carry- 
ing the  2,  we  say,  8,  13,  20,  22,  29 : 
setting  down  the  9  and  carrying 
the  2,  we  say,  9,  12,  18,  22,  30: 
and  setting  down  the  30,  we  have  the  entire  sum 
All  examples  3096.     All  the  examples  in  addition  may  be  done 

BO  solved. 

in  a  similar  manner. 


878 
421 
679 
354 
764 
3096 


Advantages 
of  reading. 


§  125.  The  advantages  of  this  method  of  read- 
ing over  spelling  are  very  great, 
lat.  stated.  1st.  The  mind  acquires  ideas  more  readily 
through  the  eye  than  through  either  of  the  other 
senses.  Hence,  if  the  mind  be  taught  to  appre- 
hend the  result  of  a  combination,  by  merely  see- 
ing its  elements,  the  process  of  arriving  at  it  is 
much  shorter  than  when  those  elements  are  pre- 
sented through  the  instrumentality  of  sound. 
Thus,  to  see  4  and  4,  and  think  8,  is  a  very  dif- 
ferent thing  from  saying,  four  and  four  are  eight. 

2d.   The  mind  operates  with  greater  rapidity 
and  certainty,  the  nearer  it  is  brought  to  the 


Sd.  stated. 


CHAP,  ir.] 


ARITHSIETIC WORDS. 


127 


ideas  which  it  is  to  apprehend  and  combine. 
Therefore,  all  unnecessary  words  load  it  and 
impede  its  operations.  Hence,  to  spell  when 
we  can  read,  is  to  fill  the  mind  with  words 
and  sounds,  instead  of  ideas. 

3d.  All  the  operations  of  arithmetic,  beyond  3d.  staled 
the  elementary  combinations,  are  performed  on 
paper ;  and  if  rapidly  and  accurately  done,  must 
be  done  through  the  eye  and  by  reading.  Hence 
the  great  importance  of  beginning  early  with  a 
method  which  must  be  acquired  before  any  con- 
siderable skill  can  be  attained  in  the  use  of 
figures. 

§  12G.  It  must  not  be  supposed  that  the  read-     Reading 

1  Til  -11  77-1  comes  attci 

ing  can  be  accomplished  until  the  spelling  has     spelling. 
first  been  learned. 

In  our  common  language,  we  first  learn  the    same  asm 

.  our  common 

alphabet,   then  we  pronounce   each  letter  m  a    language 
word,  and  finally,  we  pronounce  the  word.     We 
should  do  the  same  in  the  arithmetical  reading. 

WORDS SPELLING  AND  READING  IN  SUBTRACTION. 


§  127-  The  processes  of  spelling  and  reading  samo  piinci- 
which  we  have   explained    in   the    addition   of    insuwrao 
numbers,  may,  with  slight  modifications,  be  ap-       '^'"^ 
plied  in  subtraction.     Thus,  if  we  are  to  subtract 


128 


MATHEMATICAL     SCIENCE, 


[boc 


2  from  5,  we  say,  ordinarily,  2  from  5  leaves  3 ; 
or  2  from  5  three  remains.  Now,  the  word, 
three,  is  suggested  by  the  relation  in  which  2 
and  5  stand  to  each  other,  and  this  word  may  be 
Readings  in  read  at  oncc.     Hence,  the  reading,  in  suhtrac- 

Subtraction        .  .         .         ,  .  ,  7         >  •    7 

explained,  tion,  IS  Simply  naming  the  word,  which  expresses 
the  difference  between  the  subtrahend  and  min- 
uend. Thus,  we  may  read  each  word  of  the 
folIowinfT  one  hundred  combinations. 


READINGS 

One  from 

1 

2 

3 

4 

5 

G 

7 

8 

9 

10 

one,  &c. 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

Two  from 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

two,  &c. 

2 

0 

2 

2 

2 

2 

2 

2 

0 

2 

Three  from  3  4  5  G  78  9       10       11        12 

three,  &c.         3333333333 


Fourfrom  4  5  G  7  8  9       10       11        12       13 

four,  &c.  4444444444 


Fivefrom  5  G  7  8  9       10       11        12        13        14 

fiye,&c.    5555555555 


Six  from  six,       G        7        8        9      10      1 1      12      13      14      15 
^'^  GGGGG6GGG6 


Seven  from  7  8         9       10       11       12       13       14       15       16 

seven,  &c.        7777777777 


CHAP.  II.]  ARITHMETIC SPELLING.  129 


8  9     10     11      12     13     14     15     16     17  Eight  from 
8888888888  ''^^^^'  ^"^ 

9  10        11        12        13       14        15        16        17        18  Nine  from 

9999999999  niue,&c. 


10       11        12       13       14       15       16       17       18       19  Ten  from  ten, 

10     10     10     10     10     10     10     10     10     10  ^'^ 


§  128.   It  should  be  remarked,  that  in  subtrac- 
tion, as  well  as  in  addition,  the  spelling  of  the  speinng  pre- 

.,  1  1      •  !•  cedes  reading 

words    must  necessarily  precede   their  reading,    insubtrao- 
The  spelling  consists  in  naming  the  figures  with        '°°* 
which  the  operation  is  performed,  the  steps  of 
the  operation,  and  the  final  result.     The  reading    Reading, 
consists  in  naming  the  final  result  only. 


SPELLING    AND    READING    IN    MULTIPLICATION. 

§  129.    Spelling  in  multiplication  is  similar  to    Spelling  in 

■  ,.  .  ,  ,.   .  ,  Multiplica- 

tne  corresponding  process  in  addition  or  subtrac-       uou. 
tion.      It    is    simply   naming    the    two    elements 
which  produce  the  product ;   whilst  the  reading     Reading. 
consists    in    naming   only   the  word  which   ex- 
presses the  final  result. 

In  multiplying  each  number  from  1  to  10  by  Examples  in 
2,  we  usually  say,  two  times  1   are  2  ;  two  times     ®p""* 
2  are  4  :  two  times  3  are  6  ;  two  times  4  are  8  ; 
two  times  5  are  10;  two  times  6  are  12;  two 

9 


130  MATHEMATICAL     SCIENCE.  [bOOK  II, 

times  7  are  14;  two  times  8  are  IG;  two  times 
In  reading.    9  are  18;  two  times   10  are  20.     Whereas,  we 
should  merely  read,  and  say,  2,  4,  6,  8,  10,  12, 
14,  16,  18,  20. 

In  a  similar  manner  we  read  the  entire  mul- 
tiplication table. 

READINGS. 
Onceoneia  12       11       10       987654321 

1,  &c  2 

Twotiraesi       12     1 1     10     9     8     7     6     5     4     3     2     1 

are  2,  &c.  O 


Threetimesl  12       1 1        10       9       8       7       6       5       4       3       2       1 

are  3,  &c.  o 

Fourtimesl  12       11       10      9      8      7      6      5      4      3      2       1 

are  4,  &.c.  ^ 

Fivetimcsi  12     11     10     9     8     7     6     5     4     3     2     1 

are  5,  &c.  g 


sixtimesi        12     11     10     9     8     7     6     5     4     3     2     1 

are  six,  &.C.  Q 


Seventimes  12       1 1       10      9      8      7       6       5      4      3      2       1 

I  are  7,  &c.  7 


Eight  times  1 
are  8,  &.c. 


12     11     10     9     8     7     6     5     4     3     2     1 


CHAr.  II.]  ARITHMETIC READING.  131 


12       11        10       9       8       7       G       5       4       3       2       1  Nine  times  l 

q  are  9,  &e. 

12       11       10       9       8       7       G       5      4       3      2       1  Xentimesl 

1  n  are  10,  &c 

12       11        10       9       8       7       G       5       4       3       2       1  Eleventimes 

1  I  1  are  11,  &e. 

12       11        10       9       8       7       G       5       4       3       2       1  Xwelvetimw 

22  lnrel2,&c. 


SPELLING    AND    READING    IN    DIVISION. 

§  130.  In  all  the  cases  of  short  division,  the  inshortoivi- 

,  ,    .  1  •        1  •   I  sioii,  we  may 

quotient  may  be  read  immediately  without  nam-      read: 
ing  the  process  by  which  it  is  obtained.     Thus, 
in   dividing   the    following    numbers    by    2,    we 
merely  read  the  words  below. 

2)4       G       8       10       12       IG       18       22 

two     three     four        five  six  eight        uiue       eleven. 

In  a  similar  manner,  all  the  words,  expressing   in  aii  cases, 
the  results  in  short  division,  may  be  read. 

READINGS. 

2)2     4      G  ,    8    10    12    14    16    18    20   22    24      two  ma, 

once,  &c 

3)3      6      9    12    15    18    21   24    27    30    33    36     xhr^eina, 

once,  &c. 

4)4      8    12    16    20    24    28    32    36    40    44    48      Four  in  4, 

once,  &C. 


132  MATHEMATICAL     SCIENCE.  [bOOK  11. 


FiTeins,   5)5  10  15  20  25  30  35  40  45  50  55  60 


once,  &c. 

Six  in  6, 

6)6 

12    18 

24    30    36    42    48    54    60    66 

72 

once,  &c. 

Seven  in  7, 

7)7 

14    21 

28    35    42    49    56    63    70    77 

84 

once,  &c. 

Eight  in  8, 

8)8 

16    24 

32    40   48    56    64    72    80    88 

96 

once,  &.C. 

Nine  in  9, 

9)  9 

18  27 

36  45  54  63  72     81     90     99 

108 

once,  &c. 

Ten  in  10, 

10)10 

20  30 

40  50  60  70  80     90  100  110 

120 

once,  &c. 

Elleven  in  11, 

11)11 

22  33 

44  55  66  77  88     99  110  121 

132 

once,  &c. 

Twelve  in  12, 

12)12 

24  36 

48  60  72  84  96  108  120  132 

144 

once,  &.C. 


UNITS    INCREASING    BY    THE    SCALE    OF    TENS. 

The  idea  of  a       §  131.  The  idea  of  a  particular  number  is  ne- 
number  is    cessarilv  complex ;  for,  the  mind  naturally  asks  : 
compex.         jg^^  What  is  the  unit  or  basis  of  the  number? 
and, 

2d.   How   many    times   is    the   unit   or   basis 
taken  ? 


What  a  fig-        §  1-32.  A  figure  indicates  how  many  times  a 

Qie  indicates.         .     .  ,  _,       ,  .     ,  _  , 

unit  IS  taken,  iiiacn  or  the  ten  ngures,  however 
written,  or  however  placed,  always  expresses  as 
many  units  as  its  name  imports,  and  no  more ; 
nor  does  the  Jigure  itself  at  all  indicate  the  kind 


CHAP.  II.]     ARITHMETIC SCALE     OF     TENS.  lo3 


cf  unit.    Still,  every  number  expressed  by  one  or  Number  has 

one  for  ita 

more  figures,  has  for  its  base  either  the  abstract      basis, 
unit  one,  or  a  denominate  unit.*     If  a  denomi- 
nate unit,  its  value  or  kind  is  pointed  out  either 
by  our  common  language,  or  as  we  shall  present- 
ly see,  by  the  j)lo-ce  where  the  figure  is  written. 

The  number  of  units  which  may  be  expressed 
by  either  of  the  ten  figures,  is  indicated  by  the  Number  ex- 
name  of  the  figure.     If  the  figure  stands  alone,  single  figure. 
and  the  unit  is  not  denominated,  the  basis  of  the 
number  is  the  abstract  unit  1. 

8  133.  If  we  write  0  on  the  rijrht  of  j 

^                                                                            ""  [        10,  How  ten  is 

1,  we   have )  written. 

which  is  read  one  ten.  Here  1  still  expresses 
ONE,  but  it  is  one  ten  ;  that  is,  a  unit  ten  times 
as  great  as  the  unit  1  ;  and  this  is  called  a  unit    unit  of  the 

„     ,  J  ,  second  order. 

oi  the  second  order. 

Acrain ;  if  we  write  two  O's  on  the  ,  „     *      -. 

^  '  f       T  /-v/%  How  to  wnte 


,     100, 

right   of   1,  we    have ^  one  hundred. 

which  is  read  one  hundred.  Here  again,  1  still 
expresses  one,  but  it  is  one  hundred;  that  is,  a 
unit  ten  times  as  great  as  the  unit  one  ten,  and  a  unit  of  the 

,  11-  ,1  • .    1  third  order. 

a  hundred  tuT^es  as  c-reat  as  tlie  unit  1. 


§  134.   If  three    I's   are   written  by 


Laws — when 
IT]  figures  are 

the  side  of  each  other,  thus  .     -     .     -S  '        ^""en  by 

the  side  of 
each  other, 

*  Section  118. 


134 


MATHEMATICAL     SCIENCE. 


[book  II 


the  ideas,  expressed  in  our  common    anguage, 
are  these : 
First.  1st.  That  the  1  on  the  right,  will  either  express 

a  single  thing  denominated,  or  the  abstract  unit 
one. 
Second.  2d.  That  the  1  next  to  the  left  expresses  1  ten 

that  is,  a  unit  ten  times  as  great  as  the  first. 

Thiid.  3d.   That  the  1  still  further  to  the  left  expresses 

1  hundred ;  that  is,  a  unit  ten  times  as  great  as 

the  second,  and  one  hundred  times  as  great  as  the 

first ;  and  similarly  if  there  were  other  places. 

What  the        When  figures  are  thus  written  by  the  side  of 

eat'ablis^es    cach  Other,  the  arithmetical  language  estabUshes 

w'aen  figures  ^  relation  betwccn  the  units  of  their  places  :  that 

are  so  writ-  ^ 

len.  is,  the  unit  of  each  place,  as  we  pass  from  the 
right  hand  towards  the  left,  increases  according 
to  the  scale  of  tens.  Therefore,  by  a  law  of  the 
arithmetical  language,  the  place  of  a  figure  fixes 
its  unit. 
Scale  for        If,  then,  we  write  a  row  of  I's  as  a  scale, 

Numeration.    , . 

thus : 


13  § 

g   c  3 

,c  -2  ;s 


tH     e     S 


a;    s     ;i    o   ^ 


111,      111,      111,      111 


The  units  of 
place  deter- 
mined,    the   unit  of  each  place  is   determined,   as    well 


CHAP.  II.]      ARITHMETIC SCALE     OF     TENS.  135 


as  the  law  of  change  in  passing  from  one  place 

to  another.     If  then,  it  were  required  to  express     how  any 

number  of 

a  given  number  of  units,  of  any  order,  we  first  units  may  be 
select  from  the  arithmetical  alphabet  the  char-    ^^"^'^ 
acter  which  designates   the   number,   and   then 
write  it  in  the  place  corresponding  to  the  order. 
Thus,  to  express  three  millions,  we  write 

3000000 ; 
and  similarly  for  all  numbers. 

§  135.    It   should  be   observed,   that    a   figure  a  figure  has 

no  value  in 

being   a  character  wiiicii  represents  value,  can       itsey. 
have  no  value  in  and  of  itself     The  number  of 
things,  which  any  figure  expresses,  is  determined 
by  its  name,  as  given  in  the  arithmetical  alpha- 
bet.    The  kind  of  thing,  or  unit  of  the  figure,  is  How  the  una 
fixed  either  by  naming  it,  as  in  the  case  of  a  de-      mined, 
nominate   number,  or   by   the   place    which   the 
figure  occupies,  when  written  by  the  side  of  or 
over  other  figures. 

The    phrase    "local    value    of    a    figure,"    so  Figure, has 

no  local 

long   in   use,  is,  therefore,  without   siguificatiou      value. 

when    applied    to    a    figure:     the    term    "local 

value,"    being    applicable    to    the    unit    of   tlie    Term  ap- 
plicable to 
place,  and  not  to  the  figure  which  occupies  ihQ  unit,  of  picux. 

place. 

Federal 

§  13G.  United  States  Currency  affords  an  ex-     Money: 


136  MATHEilATICAL    SCIENCE.  [BOOK   II. 

Its  denom-  ample  of  a  series  of  denominate  imits,  increasing 

iuationa. 

according  to  the  scale  of  tens :  thus, 
-r  !-■" 

P:J    ft    P    O    S 
11111 

How  read,    may  be    read    11    thousand    1    hundred    and   11 

mills;  or,  1111  cents  and   1  mill;  or,  111  dimes 

]  cent  and  1  mill;  or,  11  dollars   1  dime  1  cent 

and  1  mill ;  or,  1  eagle  1  dollar  1  dime  1  cent 

Various  kinds  and   1   mill.      Thus,  we  may  read  the  number 

of  Reading.        .  ,         .  ,  „   .  .  ,        . 

With  either  oi  its  units  as  a  basis,  or  we  may 
name  them  all  :  thus,  1  eagle,  1  dollar,  1  dime, 
1  cent,  1  mill.  Generally,  in  Federal  Money, 
we  read  in  the  denominations  of  dollars,  cents, 
and  mills;  and  should  say,  11  dollars  11  cents 
and  1  mill. 

Examples  in       §  137.  Examples  in  reading  figures  : — 

Reading.  \        r 

istExampie.      li  wc  havc  the  hgurcs     -     -     -     -  89 

we  may  read  them  by  their  smallest 
unit,  and  say   eighty-nine ;   or,  we  may   say  8 
tens  and  9  units. 

?d.  Example.      Again,  the  figures 567 

may  be  read   by   the    smallest    unit ; 
viz.  five  hundred  and   sixty-seven ;  or  we   may 
say,  56  tens  and  7  units ;  or,  5  hundreds  6  tens 
and  7  units. 

3d.  Example.      Again,  the  number  expressed  by     -     74896 


CHAP.  II.]        ARITHMETIC VARYING     SCALES,  137 


may  be  read,  seventy-four  thousand  eight  hun-  various  read- 

.  ■  r^        •  '"S3  of  a 

dred  and  ninety-six.  Or,  it  may  be  read,  7489  number. 
tens  and  6  units ;  or,  748  hundreds  9  tens  and 
6  units ;  or,  74  thousands  8  hundreds  9  tens 
and  6  units  ;  or,  7  ten  thousands  4  thousands  8 
hundreds  9  tens  and  6  units ;  and  we  may  read 
in  a  similar  way  all  other  numbers. 

Although  we  should  teach  all  the  correct  read-     The  best 
ings  of  a  number,  we  should  not  fail  to  remark     reading. 
that  it  is  generally  most  convenient  in  practice 
to  read  by  the  lowest  unit  of  a  number.     Thus, 
in  the  numeration  table,  we  read  each  period  by  Each  period 
the  lowest  unit  of  that  period.     For  example,  in  lo'^vestunit. 
the  number 

874,967,847.047,  Example. 

we  read  874  billions  967  millions  847  thousands 
and  47. 


UMTS    INCREASING    ACCORDING    TO    VARYING    SCALES. 

§  138.  If  we  write  the  well-known  signs  of    Methods  of 
the  English  money,  and  place  1  under  each  de-   ures  having 


nomination,  we  shall  have  _,       .   , 

denommate 

£.      S.       d.      f. 
1111 


different 
nomini 
units. 


Now,  the  signs  £.  s.  d.  and^^.  fix  the  value  of     How  the 
the  unit  I  in  each  denomination;  and  they  also  unit  is  fixed. 


138  MATHEMATICAL     SCIENCE.  [bOOK   II. 

What  the    determine   the  relations  which  subsist  between 
expresses.    t^^G    different   units.     For   example,  this    simple 

ianjTuage  expresses  these  ideas  : 
The  units  of       Ist.  That  the  unit  of  the  right-hand  place  is 
e paces.    ^  farthing — of  the  place  next  to  the  left,  1  penny 
— of  the  next  place,  1  shilling — of  the  next  place, 
1  pound  ;  and 
How  the         2d.  That  4  units  of  the  lowest  denomination 
increase,     make  One  unit  of  the  next  higher;   12  of  the 
second,  one  of  the  third ;  and  20  of  the  third, 
one  of  the  fourth. 
The  units  in       If  w6  take  the  denominate  numbers  of  the 
Avoii'dupois  weight,  we  have 


Avoirdupois 
weight. 


Ton.  cwt.   qr.    lb.     oz.     dr. 

111111; 

Changes  in    in    which    the   units    increase   in    the   following 
evaueo    jj^j^j^j^g^, .    yj^.    the    second    unit,   counting   from 

the  units.  '  o 

the  right,  is  sixteen  times  as  great  as  the  first; 
the  third,  sixteen  times  as  great  as  the  second ; 
the  fourth,  twenty-five  times  as  great  as  the 
third  ;  the  fifth,  four  times  as  great  as  the  fourth ; 
and  the  sixth,  twenty  times  as  great  as  the  fifth. 
HowthescaJe  The  scale,  therefore,  for  this  class  of  denominate 

numbers  varies  according  to  the  above  laws. 

A  different        If  wc   take   any   other   class   of  denominate 

eystem!      numbers,   as  the   Troy   weight,  or  any   of  the 

systems   of  measures,   we    shall    have    different 

scales  for  the  formation  of  the  different  units. 


CHAP.  II.]      ARITHMETIC — IITTEGEAL    UKITS.  139 


But  in   all   the   formations,  we  shall   recognise  The  method 

,  f  •  r     1  1         •        •    1  °^  forming 

the  application  oi  the  same  general  principles.       ^^e  scales  tht 
There  are,  therefore,  two  general  methods  of  ^™'''°'"^ 

o  munbers. 

forming  the  different  systems   of  integral  num- 

Two  systems 

bers  from  the  unit  one.  The  first  consists  in  of  forming 
preserving  a  constant  law  of  relation  between  *°  "^^y^rT*^™" 
the  different  unities  ;  viz.  that  their  values  shall 

First  system. 

change   according   to  the  scale  of  tens.     This 
gives  the  system  of  common  numbers. 

The  second  method  consists  in  the  application   second  sys- 
of  known,  though  varying  laws  of  change  in  the 
unities.     These  changes  in  the  unities  produce  changeintha 
the  entire  system  of  denominate  numbers,  each  formi„gtho 
class  of  which  has  its  appropriate  scale,  and  the      ""'"ex- 
changes among  the  units  of  the  same  class  are 
indicated  by  the  different  steps  of  its  scale.  , 


INTEGEAL    UNITS    OF   AEITHMETIC. 

§139.   There  are  eight  classes  of  units — four  Eight  ciass- 

p  ,  T     r.  n  ■  68  of  units. 

01  number,  and  tour  of  space,  viz. 

1.  Abstract  Units;  5.  Units  of  Lines;  Abstract, 

2.  Units  of  Currency;  6.  Units  of  Surface;  currency, 

3.  Units  of  Weight;  7.  Units  of  Volumes;  Weight, 

4.  Units  of  Time;  8.  Units  of  Angles.  Time. 

First  among  the  Units  of  Arithmetic  stands 
the  simple  or  abstract  unit  1.     This  is  the  base    Abstract 

unit  one,  the 

of  all  abstract  numbers,  and  becomes  the  base,      base. 


140  MATHEMATICAL     SCIENCE.  [bO( 


The  basis  of  also,  of  all  denominate  numbers,  by  merely  na- 

denominate  .  .  .  .        ,  ,  . 

numbers;    miHg,    HI    succession,    the    particular    thmgs    to 

which  it  is  applied. 
Also,  the  ba-       It  is  also  the  basis  of  all  fractions.     Merely  as 

sis  of  all  frac- 
tions,      the  unit  1,  it  is  a  whole  which  may  be  divided 

whether  sim-  , .  ,  ^  .  .  ^ 

pieordenom-  accoi'ding  to  any  law,  lorming  every  variety  oi 

"^^'®"       fraction  ;  and  if  we  apply  it  to  a  particular  thing, 

the  fraction  becomes  denominate,   and  we  have 

expressions  for  all  conceivable  parts  of  that  thing. 


§  140.  It    has  been    remarked*    that    we  can 
Mustappre-  form  no  distiuct  apprehension  of  a  number,  un- 

heud  the 

unit.       til  we   have  a  clear  notion  of  its  unit,  and  the 

number  of  times  the  unit  is  taken.     The  unit  is 

the   great    basis.      The   utmost    care,   therefore, 

Let  its  nature  should   be   taken   to  impress    on    the   minds    of 

and  kind  be  .... 

fully  explain-  learners,  a  clear  and  distinct  idea  of  the  actual 

ed ' 

'        value  of  the  unit  of  every  number  with  which 

they  have  to  do.     If  it  be  a  number  expressing 

ow  or  a    Qyj-j.Qj^cy    Qj^g  Qj-  more  of  the  coins  should  be 

Dtimber  ex-  •'  ' 

pressingciir-  exhibited,  and  the  value  dwelt  upon;  after  which 

rency. 

distinct  notions  of  the  other  units  of  currency 
can  be  acquired  by  comparison. 

If  the  number  be   one  of  weight,  some  unit 

Exhibit  the  ^ 

anitif  itbe    should  be  exhibited,  as  one  pound,  or  one  ounce, 

of  weight; 

and  an  idea  of  its  weight  acquired  by  actually 


*  Section  110. 


CHAP.  II.]        UNITED     STATES    CIJEEEISrCT.  141 

lifting  it.  This  is  the  only  way  in  which  we 
can  learn  tlie  true  signification  of  the  terms. 

If    the    number    be    one    of    measure,   either  ^"^^^®°'^^ 

it  be  one  of 

linear,   superficial,  of  volumes   or  of  angles,  its    measm-e. 
unit  sliould  also  be  exhibited,  and  tlie  significa- 
tion of  tlie  term  expressing  it,  learned  in  the  only 
way   in    wldcli   it   can    he   learned,   througli   the 
senses,  and  by  tlie  aid  of  a  sensible  object. 

UXITED    STATES   CUERENCT. 

§  141.   The  currency  of  the  United  States  is  currency  of 
called  United  States  Currency.    Its  units  are  all     states. 
denominate,  being  1  mill,  1  cent,  1  dime,  1  dollar, 
1  eagle.     The  law  of  change,  in  passing  from  one      Law  of 

,  .  , .  ,  1         r  change  in  ths 

unit  to  another,  is  according  to  the  scale  or  tens,      unities. 
Hence,  this  system  of  numbers  may  be  treated, 

•^  J  '     How  these 

in   all   respects,  as  simple  numbers;   and  indeed '^'''"ibera may 

be  treated. 

they  are  such,  with  the  single  exception  that 
their  units  have  different  names. 

They  are  generally  read  in  the  units  of  dollars.  How  gen- 
cents,  and  mills — a  period  being  placed  after  *^'^  ^^^^  ' 
the  figure  denoting  dollars.     Thus, 

^864.849  Example. 

is  read  eight  hundred  and  sixty-four  dollars, 
eighty-four  cents,  and  nine  mills ;  and  if  there 
were  a  figure  after  the  9,  it  would  be  read  in    °''''°"''^' 

^  after  milla 

decimals  of  the  mill.      The  number  may,  how- 


142  MATHEMATICAL     SCIENCE.  [bOOK  II. 

The  number  evci',  be   Tcad   in    any  other   unit ;    as,  864849 

read  in 

vai-ious  ways,  mills ;  OT,  86484  cents  and  9  mills;  or,  8648 
dimes,  4  cents,  and  9  mills ;  or,  86  eagles,  4  dol- 
lars, 84  cents,  and  9  mills ;  and  there  are  yet 
several  other  readings. 


ENGLISH     MONEY. 

Sterling  Mo-      §  142.  The  units  of  English,  or  Sterling  Mo- 

^^'       ney,  are   1  farthing,  1  penny,   1  shilling,   and  1 

pound. 

scaieofthe        The  scale  of  this  class  of  numbers  is  a  varying 

scale.     Its  steps,  in  passing  from  the  unit  of  the 

lowest   denomination    to   the   highest,   are   four. 

How  it     twelve,  and  twenty.     For,  four  fsrthings  make 

one  penny,  twelve  pence  one  shilling,  and  twenty 

shillings  one  pound. 


unities. 


changes. 


AVOIRDUPOIS     WEIGHT. 

Units  in         §  143.  The  units  of  the  Avoirdupois  Weight 

Avoirdupois.  ,     i  ,  ,  i     ,  , 

are  1  dram,  1  ounce,  1  pound,  1  quarter,  1  hun- 
dred-weight, and  1  ton. 
Scale.  The  scale  of  this  class  of  numbers  is  a  vary- 

ing scale.    Its  steps,  in   passing  from  the  unit 
of  the  lowest  denomination   to  the  highest,  are 
sixteen,  sixteen,  twenty-five,  four,  and   twenty. 
Variation  in  ■^^^'    sixteen   drams   make   one   ounce,    sixteen 
ita  degrees,  ounccs  One  pound,  twenty-fivc  pounds  one  quar- 


CHAP.  II.]     ARITHMETIC — UKITS    OF    LEISTGTH.  143 

ter,  four  quarters  one  liundred,  and  twenty  hun- 
dreds one  ton. 

TROT    WEIGHT. 

§  144.   The  units  of  the  Troy  "Weight  are,  1     units  in 

Troy 

grain,  1  pennyweight,  1  ounce,  and  1  pound.  Weight. 

The  scale  is  a  varying  scale,  and  its  steps,  in      Scale: 
passing  from  the  unit  of  the  lowest  denomina-  its  degrees. 
tion  to  the  highest,  are  twenty-four,  twent}',  and 
twelve. 

apothecaries'    WEIGHT. 

§  145.   The  units  of  this  weight  are,  1  grain,  1     Units  in 

Apotheca- 

scruple,  1  dram,  1  ounce,  and  1  pound.  ries' Weight. 

The   scale  is  a  varying  scale.      Its   steps,  in      scaie: 
passing  from  the  unit  of  the  lowest  denomina-  its  degrees, 
tion  to  the  highest,  are  twenty,  three,,  eight,  and 
twelve. 

units   of  measure   of  space. 
§  146.   There   are  four   units  of   measure   of   ronr  units 

of  measure. 
Space,  each  differing  in  Icincl  from  the  other  three. 

They  are.  Units  of  Length,  Units  of  Surface,  Units 

of  Volume,  and  Units  of  Angular  Measure. 


units   of   length. 
§  147.   The  unit  of  length  is  used  for  measur-    Units  of 


ing  lines,   either    straight  or  curved.      It  is  a 


kugth. 


144 


MATHEMATICAL     SCIENCE.  [bOOK  If. 


The  stand-   straight  line  of  a  given  length,  and  is  often  called 

bli- 
the standard  of  the  measuren:ient. 

The  units  of  length,  generally  used  as  stand- 
ards, are  1  inch,  1  foot,  1  yard,  1  rod,  1  furlong, 
and  1  mile.  The  number  of  times  which  the 
unit,  used  as  a  standard,  is  taken,  considered  in 
connection  with  its  value,  gives  the  idea  of  the 
length  of  the  line  measured. 


What  units 
ure  taken. 


Idea  of 
length. 


UNITS     OF     SUKFACE. 


Units  of 
surface. 


What  the 

unit  of 

surface  is. 


Examples. 


Ita  connection 

with  the  unit 

of  length. 


Square  feet 

in  a 
square  yard. 


1  square  foot. 


§  148.  Units  of  surface  are  used  for  the  meas- 
urement of  the  area  or  contents  of  whatever  has 
the  two  dimensions  of  length  and  breadth.  The 
unit  of  surface  is  a  square  de- 
scribed on  the  unit  of  length 
as  a  side.  Thus,  if  the  unit 
of  length  be  1  foot,  the  corre- 
sponding unit  of  surface  will 
be  1  square  foot ;  that  is,  a  square  constructed  on 
1  foot  of  length  as  a  side. 

'  If  the  linear  unit  be  1  yard, 
the  corresponding  unit  of  sur- 
face will  be  1  square  yard.  It 
will  be  seen  from  the  figure, 
that,  although  the  linear  yard 
contains  the  linear  foot  but 
three   times,  the   square   yard 


1  yard. 


CHAP.   II. J      ARITHMETIC DUODECIMAL     UNITS.  145 

contains  the  square  foot  nine  times.  The  square  Square  rod 
rod  or  square  mile  may  also  be  used  as  the  unit  square  miie. 
of  surface. 

The  number  of  times  which  a  surface  contains     Area  or 

P  .     .  ,    contents  of  a 

its  unit  oi  measure,  is  its  area  or  contents  ;  and      gurface. 
this  number,  taken  in  connection  with  the  value 
of  the  unit,  gives  the  idea  of  its  extent. 

Besides  the  units  already  considered,  there  is  a 
special  class,  called 


DUODECIMAL     UNITS. 

§  149.  The  duodecimal  units  are  generally  used  Duodecimal 
in  board  and  timber  measure,  though  they  may  be 
used  in  all  measurements  of  surface  and  volume. 
They  are  simply  the  units  1  foot,  1  square  foot,   what  they 
and  1  cubic  foot,  divided  according  to  the  scale 
of  12. 

§  150.  It   is  proved  in   Geometry,  that  if  the -^v^jat  pnnci- 
number  of  linear  units  in  the   base  of  a  rectan-  ju'ceometry. 
gle  be  multiplied  by  the  number  of  linear  units 
in  the  breadth,  the  numerical  value  of  the  pro- 
duct will  be  equal  to  the  number  of  superficial 
units  in  the  figure. 

Knowing  this  fact,  we  often  express  it  by  say-  How  it  is  ex. 

ing,  that   "feet   multiplied  by  feet   give   square     ^'^^^^  ' 

feet,"  and  "yards  multiplied  by  yards  give  square 

10 


146 


MATHEMATICAL     SCIENCE. 


[book  11. 


ThUaconcise  yards."  But  as  feet  cannot  be  taken  jTee^  timss, 
*"'^'  .'  '  nor  yards,  yard  times,  this  language,  rightly  un- 
derstood, is  but  a  concise  form  of  expression  for 
the  principle  stated  above. 
Conclusion.  With  tliis  Understanding  of  the  language,  we 
say,  that  1  foot  in  length  multiplied  by  1  foot  in 
breadth,  gives  a  square  foot;  and  4  feet  in  length 
multiplied  by  3  feet  in  breadth,  gives  12  square 
feet. 


Kxumples  in 

the  mullipli- 

cation  of  feet 

by  feet  and 

inches. 


Generaliza- 
tion. 


Inches  by 
inches. 

How  the 

units 

change,  and 

wh-at  they 

are. 

First. 


Second. 


§  151.  If  now,  1  foot  in 
length  be  multiplied  by  1  inch 
=j2  of  ^  foot  in  breadth,  the 
product  will  be  one-twelfth 
of  a  square  foot;  that  is,  one- 
tiuelfth,  oftlie  seccmd  unit :  if  it 
be  multiplied  by  3  inches,  the  product  will  be 
three-twelfths  of  a  square  foot ;  and  similarly 
for  a  multiplier  of  any  number  of  inches. 

If,  now,  we  multiply  1  inch  by  1  inch,  the 
product  may  be  represented  by  I  square  inch : 
that  is,  iy  one-hvelfth  of  one-twelfth  of  a  square 
foot.  Hence,  tJie  units  of  this  measure  decrease 
accordi7ig  to  the  scale  of  12.     The  units  are, 

1st.  Square  feet — arising  from  multiplying  feet 
by  feet. 

2d.  Twelfths  of  square  feet — arising  from  mul- 
tiplying feet  by  inches. 


CHAP.  II.]  AEITHMETIC— UNITS,  147 

3d.  Twelfths  of  twelfths — arising  from  multi-      Third, 
plying  inches  by  inches. 

When  we  introduce  the  third  dimension,  height, 
we  have,  1   foot  being  the   nnit,  1x1x1  =  1 

.  Conclusion 

cubic  foot  ;  1  X   1  X  tV  =  T^J  cubic  foot ;  1   X  iV  X       general. 

T 2  =  T44  cubic  foot ;  and  Jg-  X  yV  X  ^V  =  ttW 
cubic  foot.  Hence,  the  units  change  by  the  scale 
of  13. 

UNITS    OF    VOLUME. 

§  152.    It    has    already    been    stated,   that    if     Units  or 

volume. 

length   be   multiplied   by   breadth,   the    product 

may  be  represented  by  units  of  surface.     It  is     what  is 

.  proved  in 

also   proved,   m    Geometry,   that   if  the   length.  Geometry  in 
breadth,  and  height  of  any  regular  figure,  of  a      them, 
square   form,   be    multiplied   together,    the   pro- 
duct  may   be   represented   by   units   of    volume     Units  or 
whose  number  is  equal  to  this  product.     Each 
unit  is  a  cube  constructed  on   tlie  linear  unit  as 
an  edge.     Thus,  if  the  linear  unit  be  1  foot,  the   Examples, 
unit  of  volume   will   be   1   cubic   foot;   that  is, 
a  cube  constructed  on   1   foot  as  an  edge;  and 
if  it  be  1  yard,  the  unit  will  be  1  cubic  yard. 

The  three  units,  viz.  the  unit  of  length,  the    The  three 
unit  of  surface,  and  the  unit  of  volume,  are  es-  tiaiiy  differ- 
sentially  different  in   kind.     The  first  is  a  line       ^"  ' 
of  a  known  length ;   the  second,  a  square  of  a  what  they 
known    side ;    and   the   third,  a  fio^ure,   called  a 


148  MATHEMATICAL     SCIENCE.  [uOOK  IT. 


Generally     cube,  of  a  known  base  and  height.     These  are 

the  units   used   in  all   kinds   of   measurement — 

Duixiccimai   excepting  only  angles,  and  the  duodecimal  sys- 

system.  . 

tem,  Avhich  has  already  been  explained. 


LIQUID     BIEASURE. 

Units  of  Li-       §  15.3.    The  units  of  Liquid   Measure   are,    1 

quid  Meas-  .  n  i  i  i 

ure.        gill,  1   pint,  1   quart,  1   gallon,  1    barrel,  1   hogs- 


head,   1    pipe,    1    tun.     The   scale   is   a  varying 
scale.       Its    steps,  in   passing  from  the  unit  of 
Howitva-    the   lowest   denomination,    are,  four,   two,   four, 
thirty-one  and  a  half,  sixty-three,  two,  and  two. 


Scale. 


nes. 


DRY     MEASURE. 

Units  ftf  Dry        §  154.  The  units  of  this  measure  are,  1  pint, 

Measure. 

1   quart,  1  peck,  1   bushel,  and  I  chaldron.     The 
Degrees  of    steps    of  the    scale,  in  passing  from  units  of  the 
lowest  denomination,  are  two,  eight,  four,   and 
thirty-six. 


Units  of  §  155.  The  units   of   Time   are,    1    second,    1 

Time.  -11 

minute,  1   hour,  1  day,  1  week,  1  month,  I  year, 
iDegreesof    and    1   century.        The   steps   of   the    scale,  in 

the  scale.  •         n  •  r     i       i  i  •         ■ 

passing  irom  units  oi  the  lowest  denomination  to 
the  highest,  are  sixty,  sixty,  twenty-four,  .seven, 
four,  twelve,  and  one  hundred. 


CHAP.  II  ]  ARITHMETIC ADVANTAGES.  149 


ANGULAK,   OE   CIRCULAR   MEASURE. 

8  156.   The  units  of  this  measure  are,  1   sec-  units  of  cir- 

cular  Meas- 

ond,  1  minute,  1   degree,  1  sign,  1   circle.     The        ure. 
steps  of  the  scale,  in  passing  from  units  of  the    Degrees  oi 

^  .         .  .  the  Scale, 

lowest  denomination  to  those  of  the  higher,  are 
sixty,  sixty,  thirty,  and  twelve. 


ADVANTAGES    OF    THE    SYSTEM    OF    UNITIES. 

§  157.  It  may  well  be  asked,  if  the  method  Acivimtases 

of  thesjsieiB 

here  adopted,  of  presenting  the  elementary  prin- 
ciples  of  arithmetic,  has  any  advantages  over 
those  now  in  general  use.  It  is  supposed  to  pos- 
sess the  following : 

1st.  The  system  of  unities  teaches  an  exact  ist.  Teaches 
analysis  of  all  numbers,  and  unfolds  to  the  mind  of  numbers: 
the  different  ways  in  which  they  are  formed  from 
the  unit  one,  as  a  basis. 

2d.  Such  an  analysis  enables  the  mind  to  form  sd. Pointsout 
a  definite  and  distinct  idea  of  every  number,  by     relation:    i 
pomting  out  the  relation  between  it  and  the  unit 
from  which  it  was  derived. 

3d.  By  presenting  constantly  to  the  mind  the  3d.  Constant 

ly  pi'piients 

idea  of  the  unit  one,  as  the  basis  of  all  numbers,   the  idea  of 
the  mind  is  insensibly  led  to  compare  this  unit      *™'^' 
with  all  the  numbers  which  flow   from  it,  and 


150 


MATHEMATICAL     SCIENCE.  [UOOK   II. 


then  it  can  the  more  easily  compare  those  num- 
bers with  each  other. 
4th.  Ex-         4th.   It    affords   a   more   satisfactory   analysis, 
^fuiiyTiie    ^iicl  a  better  nndcrstanding  of  the  four  ground 
^°"mkr"*^  rules,  and  indeed  of  all  the  operations  of  arith- 
metic, than  any  other  method  of  presenting  the 
subject. 


Primary 
bape  of 
By B tern. 


Scale. 


METEIC,  OK  FEEXCH  SYSTEM  OF  WEIGHTS 
AND   MEASURES. 

§  158.  The  primary  base,  in  this  system,  for  all 
denominations  of  weights  and  measures,  is  the 
one-ten-millionth  part  of  the  distance  from  the 
equator  to  the  pole,  measured  on  the  earth's 
surface.  It  is  called  a  Metee,  and  is  equal  to 
39.37  inches,  yery  nearly. 

The  change  from  the  base,  in  all  the  denom- 
inations, is  according  to  the  decimal  scale  of 
tens:  that  is,  the  units  increase  ten  times,  at 
each  step,  in  the  ascending  scale,  and  decrease 
ten  times,  at  each  step,  in  the  descending  scale. 


MEASURES    OF    LENGTH. 
Base,  1  metre  =  39.37  inches,  nearly. 


CHAP.   II.j  METEIC     SYSTEM.  151 


TABLE 

Ai 

Bcend^ 

ing  Scale. 

Descending 

Scale. 

A 

A 

/ 

■* 

^ 

N 

H 

^ 

B 

g3 

CD 

o 

B 
o 

H 

a3 

CD 

B 

>-> 

o 

c3 
a 

CU 

^ 

^ 

s 

w 

G 

^ 

p 

o 

s 

1 

1 

1 

1 

1 

1 

1 

1 

The  names,  in  the  ascending  scale,  are  formed    Names  ia 

the  scale. 

by  prefixing  to  tlie  base,  Metre,  the  words,  Deca 
(ten),  Hecto  (one  hundred).  Kilo  (one  thousand), 
Myria  (ten  thousand),  from  the  Greek  numerals; 
and  in  the  descending  scale,  by  prefixing  Deci 
(tenth),  Ceuti  (hundredth).  Mill  (thousandth), 
from  the  Latin  numerals. 

SQUARE     MEASURE. 

Base,  1  Are  =  Ihe  square  whose  side  is  10  metres. 
=  119.G  square  yards,  nearly. 
—  4  perches,  or  square  rods,  nearly. 

The  unit  of  surface  is  a  square  whose  side  is 
10  metres.  It  is  called  an  Are,  and  is  equal  to 
100  square  metres. 

MEASURE    OF    VOLUMES. 

Base,  1  Litre  =  the  cube  on  tlie  decimetre. 
=  G1.023378  cubic  inches. 
=  a  little  more  than  a  wine  quart. 


152  MATHEMATICAL     SCIENCE.  [BOOK   11, 

The  unit  for  the  measure  of  vohiine  is  the 
cube  whose  edge  is  one-tenth  of  the  metre — 
that  is,  a  cube  whose  edge  is  3.937  inches.  This 
cube  is  called  a  Litke,  and  is  one- thousandth 
part  of  the  cube  constructed  on  the  metre,  as 
an  edo-e. 


FOUK    GEOUND     RULES. 

System         g  159.    Let  us  take  the  two  following  examples 

applied  in 

addition,  in  Addition,  the  one  in  simple  and  the  other  in 
denominate  numbers,  and  then  analyze  the  pro- 
cess of  finding  the  sum  in  each. 


Examples. 


SEffPLE  NUMEEKS. 

DENOMINATE  NUMBBKS. 

874198 

cwt.    qr.     lb.    .  oz.      dr. 

36984 

3     3     24     15     14 

3641 

6     3     23     14       8 

914823  10     3     23     14 


Processor  In  both  examples  we  begin  by  adding  the  units 
''addiUoi""  of  the  lowest  denomination,  and  then,  we  divide 
their  sum  hy  so  many  as  mahe  one  of  the  denomi- 
nation next  higher.  We  then  set  down  the 
remainder,  and  add  the  quotient  to  the  units 
of  that  denomination.  Having  done  this,  we 
apply  a  similar  process  to  all  the  other  denomina- 
tions— the  2^rincij)le  leing  jyrecisely  the  same  in 
principle.    j^ifA   exanijjles.    We   see,  in   these   examples,  an 


CHAP.   II.]  ARITHMETIC  —  SUBTRACTION".  153 


illustration   of  a  general   principle   of  addition,  units  of  the 

•  /-    /7  7-7  7  777    Bume  kiiid 

VIZ.  that  units  of  the  same  kind  are  always  added      unite. 


together. 


§  160.   Let   us   take  two   similar  examples  in     system 

applied  in 
bubtraction.  subtraction. 


SIMPLE  NUMBEKS.  DENOMINATE  NUMBERS. 

8103  £        s.         d.   far. 

3298  6         9         7      2               Examples. 

5105  3     10       8     4 


2     18     10     2 


In  both  examples  we  begin  with  the  units  of  The  method 

of  perform- 

the  lowest  denomination,  and  as  the  number  in  iugthecx- 
tlie  subtrahend  is  greater  than  in  the  place  di- 
rectly above,  we  suppose  so  many  to  be  added 
in  the  minuend  as  make  one  unit  of  tlie  next 
higher  denomination.     We  then  make  the   sub- 
traction, and  add  1  to  the  units  of  the  subtrahend 
next  higher,  and  proceed  in  a  similar  manner, 
through  all  the  denominations.     It  is  plain  that 
the  principle  employed  is  the  same  in  both  exam-  principle  the 
pies.     Also,  that  units  of  any  denomination   in    examples, 
the  subtrahend  are  taken  from  those  of  the  same 
denomination  in  the  minuend. 


§  161.   Let  us  now  take   similar  examples   in  Muitipiica- 
Multiplication.  '^''°- 


154  MATHEMATICAL     SCIENCE.  [BOOK    II. 


SIMPLE  NTJMBESS. 

DENOMINATE  NTJMBEES. 

imples. 

87464 

5 

437320 

9 

7 

9     gr. 

G     3     15 

5 

48 

3 

2     1     15 

Method  of       lu  tliese  examplcs  we  see,  that  we  multiply,  in 

performing  .  ,  ,  „  -i-ji  ■,,•■,• 

the  exam-    successiOD,  eacii  Order  01  units  in  the  mnltipli- 

P^***-       cand  by  the  mnltiplier,  and  that  we  carry  from 

one  product  to  another,  one  for  every  so  many  as 

make  one  unit  of  the  next  higher  denomination. 

The  princi- 
ple the  same  The  ^principle  of  the  process  is  therefore  the  same 

for  all  ex-     •      n      ,  i  i 

ampius.     ^11  l^oth  examples. 


Division. 


Examples. 


§  163.   Finally,  let  us  take  two  similar  exam- 
ples in  Division. 

SIMPLE  NUMBEKS.  DENOMINATE   NUMBERS. 

3)874911  £        s.      d.  far. 

"391637  3)8       4     3     1 


3     14     8     3 


Principles  We  bcgiu,  in  both  examples,  by  dividing  the 
units  of  the  highest  denomination.  The  unit  of 
the  quotient  figure  is  the  same  as  that  of  the 
dividend.  We  write  this  figure  in  its  place,  and 
then  reduce  the  remainder  to  units  of  the  n(  xt 
The  same  as  ■^*^^^^^'  denomination.  "We  then  add  in  that  de- 
in  the  other  jjominatiou,  and  continue  the  division  through 

rules.  '  ° 

all  the  denominations  to  the  last — the  principle 
being  precisely  the  same  in  both  examples. 


CHAP,  11.]  ARITHMETIC FRACTIOVS.  155 


SECTION    II. 


FB  ACTIONAL     UNITS. 


FRACTIONAL    UNITS. SCALE    OF    TENS. 

§  1G3.    If  the  unit  1  be  divided  into  ten  equal  Fraction  one. 
parts,  each  part  is  called  one  tenth.     If  one  of     defined- 
these   tenths    be    divided  into    ten    equal    parts, 

^  ^  One 

each  part  is  called  one  hundredth.     If  one  of  the    hundredth; 
hundredths  be  divided  into  ten  equal  parts,  each        Q„g 
part  is  called  one  thousandth ;  and  corresponding   thousandth. 
names  are  given  to  similar  parts,  how  far  soever    oeneraiiza- 
the  divisions  may  be  carried. 

Now,  although    the    tenths  which    arise  from  Fractions 


are 


whole 
things. 


dividing  the  unit  1,  are  but  equal  parts  of  1, 
they  are,  nevertheless,  whole  tenths,  and  in  this 
light  may  be  regarded  as  units. 

To  avoid  confusion,  in  the  use  of  terms,  we  Fractional 
shall  call  every  equal  part  of  1  a  fractional  unit. 
Hence,  tenths,  hundredths,  thousandths,  tenths 
of  thousandths,  &c.,  are  fractional  units,  each 
having  a  fixed  relation  to  the  unit  1,  from  which 
it  was  derived. 


156  MATHEMATICAL     SCIENCE.  [boOK  IT. 

Fractional         §  164.  Adopting    a   similar  language    to   that 

units  of  the 

first  Older;    used  in  integral  numbci's,  we  call  the  tenths,  frac- 
der  &c.      tional  units  of  the  fust  ordei^ ;  the  hundredths, 
fractional    units  of  the  second  order ;  the  thou- 
sandths, fractional  units  of  the  third  order ;  and 
so  on  for  the  subsequent  divisions. 
Lan<'uac'e  for      ^^  there   any  arithmetical  language  by  which 
fractional     ^j^ggg  fractional  units  may  be  expressed  ?     The 

units.  •'  '■ 

decimal  point,  which  is  merely  a  dot,  or  period. 
What  it  fixes,  indicates  the  division  of  the  unit  1,  according  to 

the  scale  of  tens.  By  the  arithmetical  language, 
Names  of  the  the  uijit  of  the  place  next  the  point,  on  the  right, 

places. 

IS  1  tenth ;    that   of  the   second    place,   1    hun- 
dredth ;  that   of  the  third,  1  thousandth  ;  that  of 
the  fourth,    1    ten   thousandth ;    and    so   on   for 
places  still  to  the  right. 
Scale.  The  scale  for  decimals,'  therefore,  is 

.111111111,  &c.; 

in  which  the  value  of  the  unit  of  each  place  is 
known  as  soon  as  we  have  learned  the  significa- 
tion of  the  language. 

If,  therefore,  we  wish  to  express  any  of  the 
parts  into  which  the  unit  1  may  be  divided,  ac- 
cording to  the  scale   of  tens,  we  have  simply  to 

Any  decimal  ~  . 

number  may  sclcct  from    the    alphabet,    the    figure    that   will 

be  expressed 

by  this  scale,  exprcss  thc  numbev  of  parts,  and  then  write  it  in 


CIIAr.   II.]  ARITHMETIC FRACTIONS.  157 

the  place  corresponding  to  i\\Q  order  of  the  unit,   where  any 

fitjure  is 

Thus,  to  express  four  tenths,  three  thousandths,     Avritten. 
eight    ten-thousandths,    and    six    milhonths,    we 
write 

.403806  ;  H^ample. 

and    similarly,   for    any    decimal   which    can  be 
named. 

§  165.  It  should  be  observed  that  while  the 
units  of  place  decrease,  according  to  the  scale  of 
tens,  from  left  to  right,  they  increase  according  The  units  in- 

■i       r  •    1  ,    f.  rn    •      •  cicuse  from 

to  the  same  scale,  from  right  to  left.  This  is  the  ngiit  to  left 
same  law  of  increase  as  that  which  connects  the 
units  of  place  in  simple  nu?nbers.  Hence,  simple  consequence 
numbers  and  decimals  beino;  formed  according  to 
the  same  law,  may  be  written  by  the  side  of  each 
other  and  treated  as  a  single  number,  by  merely 
preserving  the  separating  or  decimal  point. 
Thus,  8974  and  .67046  may  be  written 

8974.67046  ;  Example. 

since  ten  units,  in  the  place  of  tenths,  make  the 
unit  one  in  the  place  next  to  the  left. 


FRACTIONAL    tTNITS    IN    GffNERAL. 

§  1G6.  If  the  unit  1  be  divided  into  two  equal      AhaK 
parts,  each  part  is  called  a  half.     If  it  be  divided 


158 


MATHEMATICAL     SCIENCE.  [bOOK  II. 


A  third,  into  three  equal  parts,  each  part  is  called  a  third : 
if  it  be  divided  into  four  equal  parts,  each  part  is 
called  a  fourth  :  if  into  five  equal  parts,  each 
part  is  called  a  fifth ;  and  if  into  any  number  of 
equal  parts,  a  name  is  given  corresponding  to  the 
number  of  parts. 

Now,  although  these  halves,  thirds,  fourths, 
fifths,  &c.,  are  each  but  parts  of  the  unit  1,  they 
are,    nevertheless,    in    themselves,    whole    things. 

Examples.  That  is,  a  half  is  a  whole  half;  a  third,  a  whole 
third ;  a  fourth,  a  whole  fourth  ;  and  the  same 
for  any  other  equal  part  of  1.  In  this  sense, 
therefore,  they  are  units,  and  we  call  them  frac- 

Haveaieia-  tional  uuits.     Each  is  an  exact  part  of  the  unit 


A  fourth. 
A  fifth. 

Generally. 


These  units 

are  whole 

things. 


tion  to  unity, 


1,  and  has  a  fixed  relation  to  it. 


Language  for 
fractions. 


To  express 

the  number 

of  equal 

parts. 


§  1G7.  Is  there  any  arithmetical  language  by 
which  these  fractional  units  can  be  expressed  ? 

The  bar,  written  at  the  right,  is  the 
sign  which  denotes  the  division  of  the 
ui'it  1  into  any  number  of  equal  parts. 

If  we  wish  to  express  the  number  of  equal 
parts  into  which  it  is  divided,  as  9,  for 
example,  we  simply  write  the  9  under 
the  bar,  and  then  the  phrase  means,  that  some 
thing  regarded  as  a  whole,  has  been  divided  into 
9  equal  parts. 


9 


CHAP.   II. J  ARITHMETIC FRACTIONS. 


159 


If,  now,  we  wish  to  express  any 
number  of  these  fractional  units,  as  7, 
for  example,  we  place  the  7  above  the 
line,  and  read,  seven  ninths. 


To  show  how 

many  are 

la  lien. 


§  168.  It  was  observed,*  that  two  things  are 
necessary  to  the  clear  apprehension  of  an  inte- 
gral number. 

1st.  A  distinct  apprehension  of  the  unit  which 
forms  the  basis  of  the  number ;  and, 

2dly.  A  distinct  apprehension  of  the  number 
of  times  which  that  unit  is  taken. 

Three  things  are  necessary  to  the  distinct  ap- 
prehension of  the  value  of  any  fraction,  either 
decimal  or  vulgar. 

1st.  We  must  know  the  unit,  or  whole  thing, 
from  which  the  fraction  was  derived ; 

2d.  We  must  know  into  how  many  equal  parts 
that  unit  is  divided  ;  and, 

3dly.  We  must  know  how  many  such  parts 
are  taken  in  the  expression. 

The  unit  from  which  the  fraction  is  derived, 
is  called  the  unit  of  the  fraction  ;  and  one  of 
the  equal  parts  is  called,  i\\Q  fractional  unit. 

For  example,  to  apprehend   the  value  of  the 


Two  things 

necessary  to 

apprehend  a 

niunber. 

First 


Second 


Three  things 

necessary  to 

apprehend  a 

fiaction. 


Second. 


Third. 


Hnit  of  tho 
fraction — ef 
the  expres- 
sion. 


*  Section  117. 


160 


MAT  II  EM  vriCAL     SCIENCE. 


[book    II. 


viTiatwe     fraction  f  of  a  pound  avoirdupois,  or  f  Z5. ;    we 

raust  know. 

must  know, 


First. 
Second. 


Unit  when 
not  named. 


1st.  What  is  meant  by  a  pound  ; 

2d.  That  it  has  been  divided  into  seven  equal 
parts  ;  and, 

3d.  That  three  of  those  parts  are  taken. 

In  the  above  fraction,  1  pound  is  the  unit  of 
the  fraction  ;  one-seventh  of  a  pound,  tlie  frac- 
tional unit;  and  3  denotes  that  three  fractional 
units  are  taken. 

If  the  unit  of  a  fraction  be  not  named,  it  is 
taken  to  be  the  abstract  unit  1. 


ADVANTAGES     OF     FRACTIONAL      UNITS. 


Every  equal       §109.   By  Considering  cvcrj  cqual  part  of  uni- 

part  of  one,  a 

unit.  ty  as  a  unit  in  itself,  having  a  certain  relation  to 
the  unit  1,  the  mind  is  led  to  analyze  a  frac- 
tion, and  thus  to  apprehend  its  precise  significa- 
tion. 

Under  this  searching  analysis,  the  mind  at 
ones  seizes  on  the  unit  of  the  fraction  as  the 
principal  base.  It  then  looks  at  the  value  of 
each  part.  It  then  inquires  how  many  such 
parts  are  taken. 
Equal  units.      It  having  bccn  shown  that  equal  integral  units 

whetlipr  in- 
tegral or  frac-  can   alone  be   added,  it  is  readily  seen  that  the 


Advantages 

of  the 

analysis. 


CHAP.   II  ]  ARITHMETIC ADVANTAGES.  161 

/ 

same    principle    is    equally    applicable    to    frac-    tionai,  can 

,  .  ,        ,  ....  ,  alone  be 

tional    units ;    and    then    the    inquiry  is    made :      a^de*!. 
What  is  necessary  in  order  to  make  such  units 
equal  ? 

It  is  seen  at  once,  that  two  things  are  neces-    ^wo  things 

necessary  for 

sary :  addition. 
1st.  That  they  be  parts  of  the  same  unit ;  and,       ^'^^• 

2d.  That  they  be   like  parts ;  in  other  words,  second. 

they    must    be  of  the    same   denomination,   and 

have  a  common  denominator. 

In  regard  to  Decimal    Fractions,   all    that    is  Decimal 

Fractions, 

necessary,  is  to  observe  that  units  of  the  same 
value  are  added  to  each  other,  and  when  the 
figures  expressing  them  are  written  down,  they 
should  always  be  placed  in  the  same  column. 


S  170.  The  great  difficulty  in  the  management  D'fflc""yin 

the  inanage- 

of  fractions,  consists  in   comparing    them   with  ment  of  frao 

tions, 

each  other,  instead  of  constantly  comparing  them 
with  the  unit  from  Avhich  they  are  derived. 
By  consideriijg  them  as  entire  things,  having  a 
fixed  reliition  to  the  unit  which  is  their  base,  obviated. 
they  can  be  compared  as  readily  as  integral  num- 
bers; for,  the  mind  is  never  at  a  loss  when  it 
apprehends  the  unit,  the  parts  into  which  it  is 

Eeasons  for 

divided,    and   the    number   of    parts   which   are  greater  eim- 

plicity  in 

taken.     The  only  reasons  why  we  apprehend  and    integers. 

11 


163  MATHEMATICAL     SCIENCE.  [BOOK    II. 

lianclle  integral  numbers  more  readily  than  frac- 
tions, are, 
First.  1st.  Because   the    unit    forming    the    base    is 

always  kept  in  view ;  and. 
Second.  2d.  Because,  in  integral  numbers,  we  have 
been  taught  to  trace  constantly  the  connection 
between  the  itnit  and  the  numbers  which  come 
from  it ;  while  in  the  methods  of  treating  frac- 
tions, these  important  considerations  have  beer, 
neglected. 


SECTION  III. 


PROPOKTION     AND     RATIO, 


Proportion        g  171.  PROPORTION  cxprcsses  the  relation  which 

defined. 

one  number  bears  to  another,  Vv^ith  respect  to  its 
being;  greater  or  less. 
Two  ways  of       Two  numbcrs  may  be  compared,  the  one  Avith 

comparing. 

the  other,  in  two  ways  : 
isi  method.        Ist.  With  rcspcct  to  their   difference,   called 

Arithmetical  Proportion ;  and, 
sdinothod.        2d.  With    respect    to    their    quotient,    called 

Geometrical  Proportion, 


CHAP.  II. J  ARITHMETIC PROPORTION.  163 

Thus,  if  we  compare  the  numbers  1  and  8,   Exami.ieof 
by  their  difference,  we  find  that  the  second  ex-    p,'.„py,.'tiy„; 
ceeds  the  first  by  7  :  hence,  their  difference  7, 
is  the  measure  of  their  arithmetical  proportion, 

Arithmetical 

and  is  called,  in  the  old  books,  their  arithmetical      Ratia 
ratio. 

If  we    compare   the   same    numbers    by   their  Example  of 
quotient,  we  find  that  the  second  contains   the  ^eometncaj 

'  Proportion, 

first  8  times  :  hence,  8  is  the  measure  of  their 
geometrical  proportion,  and   is  called  their  geo-        '^"°" 
metrical  ratio* 

S  172.  The  two  numbers  which  are  thus  com- 

°  Terms. 

pared,  are  called  terms.     The  first  is  called  the   Antecedent 

antecedent,  and  the  second  the  consequent.  consequent. 

In   comparing  numbers  with   respect  to  their  comparison 

j.fv.  ^,  i-  •  1  1      ■  by  difference 

dirlerence,  the  question  is,  now  much  is  one 
greater  than  the  other  ?  Their  difference  affords 
the  true  answer,  and  is  the  measure  of  their  pro- 
portion. 

In   comparing  numbers  with  respect  to  their  comparison 

,  .  .        ,  .  .  by  ([uotieHt 

quotient,  the  question  is,  now  many  times  is  one 
greater  or  less  than  the  other  ?  Their  quotient 
or  ratio,  is  the  true  answer,  and  is  the  measure 


*  The  term  ratio,  as  now  generally  used,  means  the  quo- 
tient arising  from  dividing  one  number  by  another.  We 
shall  use  it  onlv  in  this  sense. 


164  MATHEMATICAL     SCIENCE.  [boOKII. 

Example  by  of  their   proportion.      Ten,    for   exam])!©,    is    9 

difference.  ,  .  „  ,  , 

greater  than  1,  ii  we  compare  the  numbers  one 

and  ten  by  their  difference.     But  if  we  compare 

By  quotient,  them  by  their  quotient,   ten    is   said  to  be   ten 

"Ten times."  times  as  great — the  language  "ten  times"  having 

reference  to  the  quotient,  which  is  always  taken 

as   the    measure    of  the   relative    value    of  two 

Examples  of  numbers    so    compared.      Thus,   when    we   say, 

thisuseoftiie  ^j^^^  ^j^^  units  of  our  common  system  of  numbers 

term.  •' 

increase  in  a  tenfold  ratio,  we  mean  that  they  so 
increase  that  each  succeeding  unit  shall  contain 
the  preceding  one  ten  times.  This  is  a  conven- 
couveoicnt  ie^t  language  to  express  a  particular  relation  of 
language.  ^^^  numbci's,  and  is  perfectly  correct,  when 
used  in  conformity  to  an  accurate  definition. 

In  what  §  1*^3.  All  authors  agree,  that  the  measure  of 

g/    the  geometrical  proportion,  between   two  num- 
bers, is   their  ratio  ;  but  they  are  by  no  means 
jjj^ijjjt  ^jgg^.  unanimous,   nor  does  each   always    agree   with 
^'^^^       himself,  in  the  manner  of  determining  this  ratio. 
Some  determine  it,  by  dividing  the  first  term  by 
Different  me-  the  second ;  others,  by  dividing  the  second  term 
by  the  first.*     All  agree,  that  the  standard,  vvhat- 

^^Uutilard  the 

jirisor.      ever  it  may  be,  should  be  made  the  divisor. 

*  The  Encyclopedia  Metropolitana,  a  work  distinguished 
by  ihe  excellence  of  its  scientific  articles,  adopts  the  lattei 
method. 


CHAP,   II.]  ARITHMETIC RATIO.  165 

This  leads  us  to  inquire,  whether  the   mind  what  is  the 
fixes  most  readily  on  the  first  or  second  number 
as  a  standard ;  that  is,  whether  its  tendency  is 
to  regard  the  second  number  as  arising  from  the 
first,  or  the  first  as  arising  from  the  second. 

§  174.   All    our    ideas    of   numbers    begin    at     Oi-iginof 

numbeis. 

one.*     This    is    the    starting-point.     We    con- 
ceive of  a  number  only   by  measuring  it   with  now  we  cob 
one,  as  a   standard.     One   is   primarily   in   the    '^u^^er' 
mind  before  we    acquire  an  idea   of  any  other 
number.     Hence,   then,   the    comparison  begins   where  the 
at  one,  which  is  the  standard  or  unit,  and  all    '=°'"p^'^°" 

begins. 

Other  numbers  are  measured  by  it.  When,  there- 
fore, we  inquire  what  is  the  relation  of  one  to 
any  other  number,  as  eight,   the  idea  presented     „    ._, 

'  '  &      '  r  The  idea 

is,  how  many  times  does  eight  contain  the  stand-    presented. 
ard  ? 

We  measure  by  this  standard,  and  the  ratio  is     standai'd. 

Ratio. 

the  result  of  the  measurement.     In  this  view  of 

the  case,  the  standard  should  be  the  first  number  ^'^^^  ^^^^ 

should  be, 

named,  and  the  ratio,  the  quotient  of  the  second 
number  divided  by  the  first.     Thus,  the  ratio  of 
2  to  6  would  be  expressed  by  3,  three  benig  the    Example, 
number  of  times  which  0  contains  2. 


♦Section  111. 
11 


106  MATHEiMATICAL     SCIENCE.  [bOOK  II 

Other  reasons       §  175.  The  rcasoii  fof   adopting  this  method 

for  this  me-  .  .,,  -n       ^  t 

thodofoom-  of  comparison  will  appear  still  stronger,  it  we 
parison.  ^^j,^  fractional  numbers.  Thus,  if  we  seek  the 
relation  between  one  and  one-half,  the  mind  im- 
mediately looks  to  the  part  which  one-half  is  of 

Comparison  o^g^  and  this  is  determined  by  dividing  one-half 

of  unity  with 

fractions,  by  1 ;  that  is,  by  dividing  the  second  by  the 
first :  whereas,  if  we  adopt  the  other  method, 
we  divide  our  standard,  and  find  a  quotient  2. 

Geometrical        §  176.  It  may  be  proper  here  to  observe,  that 

proportion.  ,  .,        ,  •       i  •        ;>   •  i 

while  the  term  "  geometrical  proportion  is  used 
to  express  the  relation  of  two  numbers,  com- 
Ageometri-  pared  by  their  ratio,  the  term,  "  a  geometrical 
tion  defined,  proportiou,"  is  applied  to  four  numbers,  in  which 
the  ratio  of  the  first  to  the  second  is  the  same  as 
that  of  the  third  to  the  fourth.     Thus, 

Example.  2  :  4  ::  6  :   12, 

is  a  geometrical  proportion,  of  which  the  ratio 
is  2. 

Further  ad-        §  177.  We   will   uow   State   soinc  further  ad- 
\nna-es.     ^.^^^^^ggg  ^yhJcJ^  result  from  regarding  the  ratio 

as  the  quotient  of  the  second  term  divided  by 
the  first. 
Questions  in       Every  question  in  the   Rule  of  Three  is  a 

the  Kule  of  ,  .  •  i  i  i 

Three-      geometrical  proportion,  excepting  only,  that  the 


CHAP.  II. J  ARITHMEnC RATIO.  167 

last  term  is  wanting.     When  that  term  is  found,  Their  nature. 

the   geometrical    proportion    becomes    complete. 

In  all  such  proportions,  the  first  term  is  used  as 

the  divisor.     Further,  for  every  question  in  the 

Rule  of  Three,  we  have  this  ■  clear  and  simple 

solution :    viz.   that,   the   unknown   term  or  an-  how  solved. 

swer,  is   equal   to  the  third  term  multiplied  by 

the  ratio  of  the  first  two.     This  simple  rule,  for 

findino;  the  fourth  term,  cannot  be  given,  unless  Thisnuede- 

^  ^  pends  on  the 

we  define  ratio  to  be  the  quotient  of  the  second  definition  oi 

Ratio. 

term  divided  by  the  first.  Convenience,  there- 
fore, as  well  as  general  analogy,  indicates  this  as 
the  proper  definition  of  the  term  ratio. 

§178.  Again,   all    authors,   so    far    as  I    have   xhisdeflm- 

.  ^  .  i-iii','  '"3"  of  ratio  is 

consulted   them,  are  uniform  m  then'  definition    ^^^  ^^  ^^ 
of  the  ratio  of  a  geometrical   progression :   viz.    ^ne  ^T 
that  it  is  the  quotient  which  arises  from  divid- 
ing the  second  term  by  the  first,  or  any  other 
term  by  the  preceding  one.     For  example,  in 
the  progression 

2  :  4  :  8  :   IG  :  32  :  64,  &c.,  Example: 

all  concur  that  the  ratio  is  2 ;  that  is,  that  it  is     in  which 
the  quotient  which  arises  from  dividing  the  sec-       agree. 
ond  term  by  the  first :  or  any  other  term  by  the 
preceding  term.     But  a  geometrical  progression 
differs   from   a   geometrical   proportion  only  in 


168  MATHEMATICAL     SCIENCE.  [BOOK   11, 

The  same     this  :  ill  the  former,  the  ratio  of  any  two  terms 

should  take     .  i  -i      •         i        i 

piacemevery  IS  the  sume ;  while  m  the  latter,  the  ratio  of  the 
foTthey  are    ^"'^^  ^^^  secoiid  is  different  from  that  of  the  sec- 
uii  the  same,   ^^-^j  ^j^^  third.     There  is,  therefore,  no  essential 
difference  in  the  two  proportions. 

Why,  then,  should  we  say  that  in  the  propor- 
tion 

2  :  4   ::  G  :   12, 


E-vampIes. 


the  ratio  is  the  quotient  of  the  first  term  divided 
by  the  second ;  while  in  the  progression 

2  :  4  :  8  :   16  :  32  :  64,  &c., 


the  ratio  is  defined  to  be  the  quotient  of  the  sec- 
ond term  divided  by  the  first,  or  of  any  term  di- 
vided by  the  preceding  term  ? 

Wherein  As  far  as  I  havc  examined,  all  the  authors 
who  have  defined  the  ratio  of  two  numbers  to 
be  the  quotient  of  the  first  divided  by  the  sec- 
ond, have  departed  from  that  definition  in  the 
case  of  a  geometrical  progression.     They  have 

How  used  there  used  the  word  ratio,  to  express  the  quo- 
'^**'"'  tient  of  the  second  term  divided  by  the  first, 
and  this  without  any  explanation  of  a  change 
in  the  definition. 

other  in-  Most  of  them  havc  also  departed  from  theii 
definition,   in  informing    us    that  "  numbers    in- 


authoi'S 
have  depart- 
ed from  their 
defluitions : 


stances  in 
which  the 

jefinitionof  urease  from  right  to  left  in  a  tenfold  ratio,"  in 


CHAP.  II.]  ARITHMETIC PROPORTION.  169 

which  the  term  ratio  is  used  to  denote  the  quo-  Ratio  is  not 

tient  of  the  second  number  divided  by  the  first. 

The  definition   of  ratio   is   thus  departed   from, 

and    the    idea   of   it   becomes    confused.      Such   consequen- 
ces. 
discrepancies    cannot    but    introduce    confusion 

into    the    minds    of  learners.      The    same    term 

should  always  be  used  in  the  same  sense,  and 

have   but  a  single    signification.     Science    does  what  science 

I  T    1  r  I  •  demands, 

not  permit  the  slightest  departure  from  this  rule. 
I  have,  therefore,  adopted  but  a  single  significa- 
tion of  ratio,  and  have  chosen  that  one  to  which   xhedeflm- 

.  .      tion  adopted 

all  authors,  so  lar  as  1  know,  have  given  their 
sanction  ;  although  some,  it  is  true,  have  also 
used  it  in  a  different  sense. 


§  179.  One  important  remark  on  the  subject    importam 

r  •         ■  1  1  T     •        1  •  Remark. 

01  proportion  is  yet  to  be  made.     It  is  this  : 

Any  two  numbers  which  are  compared  togeth-     Numbers 

compared 

er,  cither  hij  their   difference   or  quotient,   must    must  be  of 

I  ^      ,  7  •      7         7  •  I  •   ;  '•'^^  same 

be  of   the  same   knid:  that  is,   they  must  either       i^inj^ 
have  tlte  same  unit,  as  a  base,  or  be  susceptible 
of  reduction  to  the  same  unit. 

For  example,  we  can  compare  2  pounds  with    Examples 

1  ^      •        ^•  rf  •  ^  i      i      •         relating  to 

G  pounds  :  their  difference  is  4  pounds,  and  their  Arithmetic!^,' 
ratio  ii.  the  abstract  number  3.     We   can  also  i-icai  Propor- 
compare  2  feet  with  8  yards  :  for,  although  the       '^'""' 
unit  1  foot  is  different  from  the  unit  1  yard,  still 
8  yards  are  equal  to  24  feet.     Hence,  the  differ- 


170  MATHEMATICAL     SCIENCE.  [bOOK  II. 

ence  of  the  numbers  is  22  feet,  and  their  ratio 
the  abstract  number  12. 
Numbers         On  the  Other  hand,  we  cannot  compare  2  dol- 

with  different 

units  cannot  lars  with  2  yai'ds  of  cloth,  for  they  are  quantities 

be  com paie J.       ,     . 

oi  different  kinds,  not  being  susceptible  of  reduc- 
tion to  a  common  unit. 
Abstract        Abstract  numbers  may  always  be  compared, 

Qumbers  may 

be  corapaj-ed.  sincc  they  have  a  common  unit  1. 


SECTION  IV. 


APPLICATIONS    OF    THE    SCIENCE    OF    ARITHMETIC". 

§  180.  Arithmetic    is    both  a  science  and  an 
Arithmetic:  art.     It  is  a  sciencc  in    all   that  relates  to  the 

In  what  a  .  .  „  , 

science,      properties,    laws,    and   proportions    oi    numbers. 

The  science  is  a  collection  of  those  connected 

Science  de-   processcs  whicli  dcvclop  and  make  known   the 

fined. 

laws  that  regulate  and  govern  all  the  operations 
performed  on  numbers. 


science  per- 
forms. 


§  181.  Arithmetic  is  an  art,  in  this  :  the  sci- 
ence lays  open  the  properties  and  laws  of  num- 
bers, and  furnishes  certain  principles  from  which 


CHAP.  II.]  ARITHMETIC APPLICATIONb  171 


practical  and  useful  rules  are  formed,  applicable 
in  the  mechanic  arts  and  in  business  transac- 
tions. The  art  of  Arithmetic  consists  in  the  in  what  the 
judicious  and  ski.ful  application  of  the  princi- 
ples of  the  science ;  and  the  rules  contain  the 
directions  for  such  application. 

§  1S3.   In  explaining  the  science  of  Arithmetic,  in  explaining 
great  care  should  be  taken  that  the  analysis  of  ^hltlilcessa 
every  question,  and  the  reasoning  by  which  the        ^^' 
orinciples  are  proved,  be  made  according  to  the 
strictest  rules  of  mathematical  loG;ic. 

Every  principle  should  be  laid  down  and  ex-    how  each 
plained,  not  only  with  reference  to  its  subsequent    s'j'oullfi,''^ 
use  and  application  in  arithmetic,  but  also,  with      ^'^'^^'^ 
reference  to  its  connection  with  the  entire  mathe- 
matical science — of  which,  arithmetic  is  the  ele- 
mentary branch. 

§  183.  That   analysis   of  questions,  therefore,       what 

1  .     •  J        '^L  iV  questions  aj» 

where  cost  is  compared  with  quantity,  or  quan-  ^^^^^ 
tity  with  cost,  and  which  leads  the  mind  of  the 
learner  to  suppose  that  a  ratio  exists  between 
quantities  that  have  not  a  common  unit,  is,  with- 
out explanation,  certainly  faulty  as  a  process  of 
science. 

For  example  :  if  two  yards  of  cloth  cost  4  dol- 

Example. 

lars,  what  will  6  vards  cost  at  the  same  rate  ? 


172  MATHEMATICAL     SCIENCE.  [boOKU 

Analysis:  Analysts. — Two  yards  of  cloth  will  cost  twice 
as  much  as  1  yard  :  therefore,  if  two  yards  of 
cloth  cost  4  dollars,  1  yard  will  cost  2  dollars. 
Again  :  if  1  yard  of  cloth  cost  2  dollars,  G  yards, 
being  six  times  as  much,  will  cost  six  times  two 
dollars,  or  12  dollars. 
Satisfactory        Now,  this  analysis   is  perfectly  satisfactory  to 

to  a  child.  i  -i  i        tt  •  •  i      ■  i 

a  child.  He  perceives  a  certain  relation  between 
2  yards  and  4  dollars,  and  between  6  yards  and 
12  dollars :  indeed,  in  his  mind,  he  compares 
these  numbers  together,  and  is  perfectly  satisfied 
with  the  result  of  the  comparison. 

Advancing  in  his  mathematical    course,  how- 
ever, he  soon   comes  to   the   subject  of  propor- 
tions,   treated    as    a   science.     He    there    finds, 
Reason  why  greatly  to   his  surprise,  that  he  cannot  compare 

it  is  defective.  i  •    i      i  ^■  ly 

together  numbers  which  have  different  units ; 
and  that  his  antecedent  and  consequent  must  be 
of  the  same  kind.  He  thus  learns  that  the  whole 
system  of  analysis,  based  on  the  above  method  of 
comparison,  is  not  in  accordance  with  the  prin- 
ciples of  science. 
True  What,  then,  is  the  true  analysis  ?     It  is  this  : 

mialysis :  i  i'       i      i      i      •  • 

G  yards  of  cloth  being  3  times  as  great  as  2 
yards,  will  cost  three  times  as  much  :  but  2  yards 
cost  4  dollars  ;  hence,  6  yards  will  cost  3  times 
4,   or   12   dollars.     If  this    last    analysis    be   not 

More  acien-  •' 

*^'^'=-        as  simple  as  the  first,  it  is  certainly  moie  strictly 


CHAP,   ir.J  ARITHMETIC APPLICATIONS.  173 


scientific  ;    and  when  once  learned,  can  be  ap-         its 
plied  through  the  whole  range  of  mathematical 
science. 


§  184.  There  is  yet  another  view  of  this  ques-   Reasons  in 
tion  which   removes,   to  a  great  degree,  if  not  aret  analysis 
entirely,  the  objections  to  the  first  analysis.    It  is 
this : 

The  proportion  between  1  yard  of  cloth  and 
its  cost,  two  dollars,  cannot,  it  is  true,  as  the 
units  are  now  expressed,  be  measured  by  a  ratio, 
according  to  the  mathematical  definition  of  a 
ratio.  Still,  however,  between  1  and  2,  regard- 
ed as  abstract  numbers,  there  is  the  same  relation  Numbera 
existing  as  between  the  numbers  6  and  12,  also  ™"*"'«"''^' 

"  garded  as  ab 

regarded  as  abstract.     Now,  by  leaving  out  of      s"'a<=*= 

view,  for   a  moment,  the  units  of  the   numbers, 

and  finding  12  as  an  abstract  number,  and  then  The  analysis 

•  ^    • .  •  ^  1  ,     then  correct. 

assignmg  to  it  its  proper  unit,  we  have  a  correct 
analysis,  as  well  as  a  correct  result. 


§  185.  It  should  be  borne  in  mind,  that  practi-  How  the 

rules  of  aiith- 

cal  arithmetic,  or  arithmetic   as  an   art,   selects  meticare 

from  all  the  principles  of  the  science,  the  mate-  °'""''  " 
rials  for  the   construction    of  its  rules  and  the 

proofs   of  its  methods.     As   a  mere  branch  of  yy^^ 

practical  knowledge,  it  cares  nothing  about  the  v^a-cucai 

^                                             ^    '                                          ^  knowledgo 

forms  or  methods  of  investigation — ^^it  demands  <Jemanda, 


174  MATHEMATICAL    SCIENCE.  [l^OO^  ^^• 

the  fruits  of  them  all,  in  the  most  concentrated 
Best  rule  of   ^nd  practical  form.     Hence,  the  best  rule  of  art, 
which  is  the  one  most  easily  applied,  and  which 
reaches  the  result  by  the  shortest  process,  is  not 
always   constructed   after  those    methods  which 
science  employs  in  the  development  of  its  prin- 
ciples. 
Deflniiinn  of       For  example,  the  definition  of  multiplication  is, 
""tion."^'*'    ^^^^^  i^  i^  ^^^  process  of  taking  one  number,  called 
the   multiplicand,  as    many  times    as    there   are 
What  it  de-   ^^^^^  ^^^  another  called  the  multiplier.     This  defi- 
mands.      nition,  as  one  of  science,  requires  two  things, 
first.  1st.  That  the  multiplier  be  an  abstract  num- 

ber; and, 
Second.  2dly.  That  the  product,  be  a  quantity  of  the 

same  kind  as  the  multiplicand. 

These    two  principles    are    certainly   correct. 

Maybe      ^^d  relating  to  arithmetic  as  a  science,  are  uni- 

differentiy    ^.^^-gj^Uy  ^j.jjg_     jj^^.  ^^e  they  Universally  true,  in 

considered  as  •'  j  ^ 

furnishing  a  ^hg  scusc  ill  which  thev  would  be  understood  by 

nUe  of  art 

learners,  when  applied  to  arithmetic  as  a  mixed 
subject,  that  is,  a  science  and  an  art  ?  Such  an 
application  would  certainly  exclude  a  large  class 
of  practical  rules,  which  are  used  in  the  api)li- 
cations  of  arithmetic,  without  reference  to  par- 
ticular units. 
Examples  of       Yov  example,  if  we  have  feet  in  length  to  be 

sucti 

applications,  multiplied  by  feet  in  height,  we  must  exclude  the 


CHAP.  II.]  ARITHMETIC APPLICATIONS.  175 


question  as  one  to  which  arithmetic  is  not  appli- 
cable ;  or  else  we  must  multiply,  as  indeed  we 
do,  without  reference  to  the  unit,  and  then  assign 
a  proper  unit  to  the  product. 

If  we  have   a  product  arising  from  the  three    ^^^n  the 

three  factors 

factors    of  length,    breadth,  and    thickness,    the     arc  lines. 
unit  of  the  first  product  and  the  unit  of  the  final 
product,   will   not    only   be   different    from  each 
other,  but  both  of  them   will   be   different  from 
the  unit  of  the  given  numbers.     The  unit  of  the  The  diirereni 
given  numbers  will  be  a  unit  of  length,  the  unit      ™' 
of  the  first  product  will  be  a  square,  and  that,  of 
the  final  product,  a  cube. 

§  186.  Again,  if  we  wish  to   find,  by  the  best      °"'^'' 

examples, 

practical  rule,  the  cost  of  467  feet  of  boards  at 
30  cents  per  foot,  we  should  multiply  467  by 
30,  and  declare  the  cost  to  be  14010  cents,  or 
$140.10. 

Now,  as  a  question  of  science,  if  you  ask,  can    considered 
we  multiply  feet  by  cents  ?  we  answer,  certainly 
not.     If  you  again    ask,  is   the   result   obtained 
right  ?  we  answer,  yes.    If  you  ask  for  the  analy- 
sys,  we  give  you  the  following : 

1  foot  of  boards  :  467  feet  :  :  30  cents  :  Answer. 

Now,  the  ratio  of  1   foot  to  467  feet,  is  the  ab       RaUo. 
stract  number  467 ;  and  30  cents  being  multi- 


as  a  questioc 
of  science. 


176  MATHEMATICAL     SCIENCE.  [boOK    II, 

plied  by  this  number,  gives  for  the  product  14010 
cents.     But   as   the  product  of  two  numbers  is 

Product  of 

two       numerically  the  same,  whichever  number  be  used 

numbers 

as  the  multiplier,  we  know  that  467  multiplied  by 

30,  gives  the  same  number  of  units  as  30  multi- 

Tho  first  rule  pHed  by  467 :  hence,  the  first  rule  for  finding  the 

correct. 

amount  is  correct. 
Scientific  in-       §  187.  I  havc  given  these  illustrations  to  ponit 

vestigation :  ii-rr-  i.  c         •        ^-  r 

out  the  dmerence  between  a  process  oi  scientinc 
Practica      investigation  and  a  practical  rule. 

rule :  ^ 

The   first   should  always  present  the  ideas  ot 
Their  difler-  the  subjcct  iu  their  natural  order  and  connection, 
what  it  con-  while  the  other  should  point  out  the  best  way  of 
obtaining   a   desired    result.     In    the    latter,   the 
steps  of  the  process  may  not  conform  to  the  or- 
der necessary  for  the  investigation  of  principles ; 
but  the  correctness  of  the  result  must  be  suscepti- 
ble of  rigorous  proof.     Much  needless  and  un- 
,    profitable  discussion  has  arisen  on  many  of  the 

Causes  of      ^  "^ 

error.       proccsses  of  arithmetic,  from  confounding  a  princi- 
pie  of  science  with  a  rule  of  mere  application. 


CHAP.   II.] 


ARITHMETIC ORDEU. 


177 


SECTION  y. 


METHODS    OF    TEACHING    ARITHMETIC    COXSIDEEED. 


ORDER    OF    THE    SUBJECTS. 


§  188.  It  has  been  well  remarked  by  Cousin, 
the  great  French  philosopher,  that  "  As  is  the 
method  of  a  philosopher,  so  will  be  his  system ; 
and  the  adoption  of  a  method  decides  the  destiny 
of  a' philosophy." 

What  is  said  here  of  philosophy  in  general,  is 
eminently  true  of  the  philosophy  of  mathematical 
science ;  and  there  is  no  branch  of  it  to  which 
the  remark  applies,  with  greater  force,  than  to 
that  of  arithmetic.  It  is  here,  that  the  first  no- 
tions of  mathematical  science  are  acquired.  It 
is  here,  that  the  mind  wakes  up,  as  it  were,  to 
the  consciousness  of  its  reasoning  powers  Here, 
it  acquires  the  first  knowledge  of  the  abstract — 
separates,  for  the  first  time,  the  pure  ideal  from 
the  actual,  and  begins  to  reflect  and  reason  on 
pure  mental  conceptions.  It  is,  therefore,  of  the 
highest  importance  that  these  first  thoughts  be 
impressed  on  the  mind  in  their  natural  and  proper 

12 


Cousin. 

Method 

ducidea 

Philosophy. 


True  In 
science. 


Why 
important  in 
Arithmetic. 


Firet 

thoughts 
eliould  be 

righlly 
inipreti&ed. 


178  MATHEMATICAL     SCIENCE.  [UOOKI!. 

Faculties  to    ordei",  SO  US  to  Strengthen  and  cultivate,  at  the 

be  cultivated.  .  i        c  i    •  r  i  •  i  •  • 

same  time,  the  laculties  oi  apprehension,  discrim- 
ination, and  comparison,  and  also  improve  the 
yet  higher  faculty  of  logical  deduction. 

first  point:        g  189.    The    fii'st   poiut,    then,    in    framing   a 
course  of  arithmetical   instruction,  is  to  deter- 
methodof    mine  the  method  of  presenting  the  subject.     Is 
the  subject,    there  any  thing  in  the  nature  of  the  subject  it- 
self, or  the  connection  of  its  parts,  that  points 
out  the   order  in  which  these  parts  should  be 
Laws  of     studied?     Do    the   laws    of  science   demand    a 

science :  .  ,  , 

wiiat  iio     particular   order ;    or   are    the   parts    so    loosely 
th.^y require?  connected,  as  to  render  it  a  matter  of  indiffer- 
ence where  we  begin  and  where  we  end  ?     A 
review  of  the  analysis  of  the  subject  will  aid  us 
in  this  inquiry. 


iiasisofthe       §  190.    We  havc  seen*  that  the  science  of 

SCitillCG  of 

numbers,  numbci's  is  bascd  on  the  unit  1.  Indeed,  the 
In  what  the    wholc    scicncc  consists    in   developing,  explain- 

consists.     ij^g)   ^^^   illustrating    the    laws   by    which,    and 

through  which,  we  operate  on  this  unit.     There 

Four  classes  are  four  classes  of  operations  performed  on  the 

of  opera- 
tions,      unit  one. 

1st.  To  in-       1st.  To    increase    it    according    to    the    scale 

crease  the 

unit.         — — — — - 


*  Section  101. 


CHAP.  II.]      AEITHMETIC — INTEGRAL    UNITS.  179 

of  tens   forming  the   system  of  common  num- 
bers. 

2d.  To  divide  it,  in  any  way  we  please,  form-   g^j  toju. 
ing  the  decimal  and  vulgar  fractions.  ^^  ^' 

3d.  To    increase    it    according    to    the    vary-    34.  to  in- 
ing    scales,   forming   all    the   denominate    num- 
bers. 

4th.  To  compare  it  with  all  the  numbers  which  4th.  To  com- 
pare it. 
come  from  it;  and  then  those  numbers  with  each 

other.     This  embraces  proportions,  of  which  the 

Rule  of  Three  is  the  principal  branch. 

There  is  yet  a  fourth  branch  of  arithmetic;      Fourth 

viz.  the  application  of  the  principles  and  of  the 

rules  drawn    from   them,  in  the   mechanic  arts  practic.ii  ap- 

T      .         .,  ,.  .  ,.  jy     ^        •  plications; 

and    m   the   ordinary  transactions   ot    business. 
This   is   called   the    Art,   or    practical    part,   of     these  the 
Arithmetic.     (See  Arithmetical   Diagram  facing 
page  119.) 


INTEGRAL    UNITS. 
§  191.    We   begin   first   with   the   unit  1,  and    Unit  one 

increased 

increase  it  according  to  the  scale  of  tens,  form- according  to 

ing  the  common  system  of  integral  numbers.    We       tpug 

then  perform  on  these  numbers  the  operations 

of  the  five  ground  rules;    viz.   numerate  them,   operatioH^ 

add  them,  subtract    them,  multiply  and   divide  p^'""'^™*'^" 

them. 


180  MATHEMATICAL     SCIENCE.  [BOOK   11. 


FRACTIONAL    UXITS. 

§  192.   We   next   pass  to  the   second   class  of 
Divisions  of  Operations  on  the  unit  1 ;  A'iz.  tlie  divisions  of  it. 

the  unit. 

General     Here  Ave  pni'sue  tlie  most  general  method,  and 
method,     gj.gj-  (jiYi(](a  jt  arbitrarily;  that  is,  into  any  num- 
ber of  equal   parts.    We  then  observe  that  the 
Method  ac-  divisiou  of  it,  according  to  the  scale  of  tens,  is 
8caieof"ens.  ^^^t  ^  particular  case  of  the  general  law  of  divi- 
sion.     AVe   then   perfoi'm   on    all   the   fractional 
units  which   thus   arise,  every  operation  of  the 
five  eround  rules. 


DENOMINATE     UNITS. 

§  193.   Having  operated  on  the  abstract  unit  1, 

by  the  processes  of  augmentation   and  division, 

Next  in-    Ave   uext   iucrcasc   it   according    to   tlie   A'arying 

crease  it  tic-  i  ,.     i         t  •       ,  ,  i    ,  i 

cordin<,'to    scalcs  ot  the  denominate  nunibei'S,  and  tlius  pro- 
^soi'cs"     duce  the  system,  called  Denominate  or  Concrete 
Numbers;  alter  which,  we  perform  on  this  class  of 
numbers  all  the  operations  of  the  iive  ground  rules. 
Kcasonsfor      ^J   phicing   the   subjcct    of  fractions   directly 
^  ^ uons'^*'*^'  ^^^^^'  ^''6  five  gr>;und  rules,  the  two  opposite  oper- 
ations of  aggregation  and   divisi(ni  are  brought 
into  direct  contrast  with  each  other.     It  is  thus 
seen,  that  the  laws  of  change,  in  the  tAvo  systems 
of  opei-ation  on  the   unit  1,  are  the  same  with 
very  slight  modifications. 


CHAP.  II.]  AEITHMETIC  —  RATIO.  181 


This  system  of  classification,  has,  after  expe- 
rience, been  found  to  be  the  best  for  instruction. 

RATI  O, 0  R     RULE     OF     THREE. 

8  194.  Having  considered  the  two  subjects  of     subjects 

"  considered. 

integral  and  fractional  units,  we   come   next  to 

the  comparison  of  numbers  wnth  each  other. 

This  branch   of  arithmetic    develops    all    the    what  this 

.  .        r         branch  de- 
relative   properties   of  numbers,  resultmg   Irom      yeiops. 

their  inequahty. 

The  method  of  arrangement,  indicated  above,  whatiheur 

rangement 

presents  all  the  operations  of  arithmetic  ni  con-       does, 
nection  with  the  unit  1,  which  certainly  forms 
the  basis  of  the  arithmetical  science. 

Besides,  this  arrangement  draws  a  broad  line  what  u does 

....  ,     .  farther. 

between  the  science  ot  arithmetic  and  its  ap- 
plications ;  a  distinction  which  it  is  very  im- 
portant to  make.     The  separation  of  the  prin-   Theory  and 

,      .  T         •  practice 

ciples  of  a   science  from   their   applications,  so  should  be 
that   the  learner  shall   clearly  perceive  what  is  ^''P'^^''^- 
theory  and  what  practice,  is  of  the  highest  im- 
portance.    Teaching  things  separately,  teaching  Golden  rules 

,  11  1  •       •  1      •  X-  forteacliing. 

them  well,  and  pointing  out  their  connections, 
are  the  golden  rules  of  all  successful  instruc- 
tion. 


195.  I  had  supposed,  that  the  place  of  the 


182  MATHEMATICAL     SCIENCE.  [nOOK  II. 

Rule  of  Three,   among   the  branches   of  arith- 
metic, had  been  fixed  long  since.     But  several 
Differences  in  authors  of  late,  havc  placed  most  of  the  practi- 

arrangement;        i        ,  .       ,       ,     ,-  ,  .  ,  •    ■  , 

cal  subjects  bejore  this  rule — giving  precedence, 

for  example,  to  the  subjects  of  Percentage,  In- 

in  what  they  td'cst,  Discount,  Insurancc,  &c.     It  is  not  easy 

consist. 

to  discover   the   motive   of   this   change.     It   is 
Ratio  pnrt  of  certain   that  the  proportion  and   ratio  of  num- 

the  science. 

bers  are  parts  of  the  science  of  arithmetic  ;  and 
Should  pre-   the  properties   of  numbers    which    they   unfold, 

cede  applica- 

uons.  are  indispensably  necessary  to  a  clear  apprehen- 
sion of  the  principles  from  which  the  practical 
rules  are  constructed. 

We  may,  it  is  true,  explain  each  example  m 

Percentage,  Interest,  Discount,  Insurance,  &:c., 

Cannot  well   by   a   Separate   analysis.     But    this    is  a  matter 

order.       of  much  labor ;   and  besides,  does  not   conduct 

the    mind   to    any  general    principle,   on    which 

all  the  operations  depend.     Whereas,  if  the  Rule 

of  Three   be  explained,  before  entering  on  the 

Advantages   practical  subjccts,  it  is  a  great  aid  and  a  pow- 

?.,'.^ „*'.'[,'    erful    auxiliary    in    explaininsr    and    establishing 

plaining  tno  J  JT  o  a 

^""^''f      all  the  practical  rules.     If  the    Rule  of  Three 

Three.  ^ 

is    to   be    learned    at    all,    should    it   not   rather 

precede  than  follow  its   applications  ?      It  is   a 

great  point,  in  instruction,  to  lay  down  a  gen- 

The great    gj.g^j  principle,  as  early  as  possible,  and  then  con- 

principleof  ^  1      '  J  i 

instruction,    nect  witli  it  all  subordinate  operations. 


CHAP,  ir.]  >VRITH:.rETIC LANGUAGE.  183 


ARITHMETICAL     LANGUAGE, 

§  196.  We  have  seen  that  the  arithmetical  al-  Arithmetical 

■         -n  1  alphabet. 

phabet  contains  ten  characters.*  From  these 
elements  the  entire  language  is  formed;  and  we 
now  propose  to  show  in  how  simple  a  manner. 

The  names  of  the  ten  characters  are  the  first  Names  of  the 

cliaracters. 

ten  words  of  the  language.     If  the    unit   1    be 

added  to  each  of  the  numbers  from   0  to  9  in-     First  ten 

elusive,   we  find   the   first   ten    combinations   in       tions. 

arithmetic. t     If  2   be    added,   in   like   manner, 

we  have  the  second  ten  combinations ;    adding  Second  ten, 

ami  so  on  fof 

3,  gives  us  the  third  ten  combinations ;  and  so      othera. 
on,  until   we  have  reached  one   hundred  com- 
binations (page  123). 

Now,  as  we  progressed,  each  set  of  combina-  Each  setgiv. 

ing  one  addi- 
tions  introduced  one   additional  word,   and   the  tionaiword. 

results  of  all  the  combinations  are  expressed  by 

the  words  from  two  to  twenty  inclusive. 


§  197.   These  one  hundred  elementary  com-  ah  that  need 

be  commit' 

binations,  are   all   that   need   be   committed    to    ted  tome- 
memory  ;  for,  every  other  is  deduced  from  them,         °^^' 
They  are,  in  fact,  but  different  spellings  of  the 
first  nineteen  words  which  follow  one.    If  we  ex- 
tend the  words  to  one  hundred,  and  recollect  that 


*  Section  114.  f  Section  116, 


184  MATHEMATICAL     SCIENCE.  [bOOK  II. 


at  one  hundred,  we  begin  to  repeat  the  numbers, 
worJstobe  we  See  that  we  have  but  one  hundred  words  to 

remembered  i  i     r  i  i-    •  i        r       i  ti 

for  addition.   06  remembered  lor  addition;   and  ot    these,  alt 
Only  ten      obove    ten    are    dericaiice.      To    this    number, 

words  primi- 
tive,       must  of  course  be  added  the  few  words  which 

express  the  sums  of  the  hundreds,  thousands,  &c. 

Subtraction:        §  108,  In  Subtraction,  we  also  find  one  hun- 
dred  elementary   combinations;    the    results    of 
which  are  to  be  read.*     These  results,  and  all 
Number  of   the   iiumbers   employed   in    obtaining  them,   are 

words. 

expressed  by  twenty  words. 

Muitipiica-  §  199.  In  Multiplication  (the  table  being  car- 
ried to  twelve),  we  have  one  hundred  and  forty- 
four    elementary    combinations,!    and    fifty-nine 

Number  of  Separate  words  (already  known)  to  express  the 
results  of  these  combinations. 

Division:  §  200.  In  Divisiou,  also,  we  have  one  hundred 
and   forty-four    elementary    combinations,!    but 

Number  of  -^  •'  ^ 

words.      ygg  only  twelve  words  to  express  their  results. 

Four  hun- 

dred  and         o  ^q-^^   T\\\xs,  wc  havc  four  hundred  and  eigh- 

ighty-eight  "  '  » 

elementary    ty-cight  elementary  combinations.     The  results 

combina- 
tions,      of  these  combinations  are  expressed  by  one  hun- 

wordsused:  j^,^^  words  ;  viz.  nineteen  in  addition,  ten  in  sub- 

19  in  addi- 
tion'       traction,  fifty-nine  in  multiplication,  and  twelve 

10  in  subtrac 

tion, — ■ 

59inmuiti-        *  Section  127.         f  Section  129.         :t  Section  130. 
plication,  '  ^ 


CHAP.   II.]  ARITHMETIC LANGUAGE.  185 


in  division.  Of  the  nineteen  words  which  are  isin division 
employed  to  express  the  results  of  the  combina- 
tions in  addition,  eight  are  again  used  to  express 
similar  results  in  subtraction.  Of  the  fifty-nine 
which  express  the  results  of  the  combinations 
in  multiplication,  sixteen  had  been  used  to  ex- 
press similar  results  m  addition,  and  one  in 
subtraction ;  and  the  entire  twelve,  which  ex- 
press the  results  of  the  combinations  in  division, 
had  been  used  to  express  results  of  previous 
combinations.  Hence,  the  results  of  all  the  ele- 
mentary combinations,  in  the  four  ground  rules, 
are  expressed  by  sixty-three  different  words  ;  and    sixty-three 

I  11111  dilleiout 

they  are  the  only  words  employed   to  translate  words  in  aii. 
these  results  from  the  arithmetical  into  our  com- 
mon language. 

The  language   for  fractional  units   is  similar    Language 

,  -P,  r  1  '^s  same  for 

in  every  'particular.     By  means  oi   a  language     fractions. 
thus  formed  we  deduce  every  principle  in   the 
science  of  numbers. 

§  202.  Expressing  these  ideas  and  their  com- 
binations by  figures,  gives  rise  to  the  language  Language  of 

f.         .,  .  -pi         I  •iri'i  aiitlimetic: 

OI  arithmetic.     r5y  the  aid  oi  this  language  we 

not   only  unfold    the   principles   of   the   science,  its  value  and 

but    are    enabled    to    apply    these    principles  to 

every  question  of  a  practical  nature,  involving 

the  use  of  fio-ures. 


186 


MATHEMATICAL     SCIENCE, 


[book  II. 


But  few 
combinations 

whicli 
change  the 
sigaificatiun 
of  the  figures. 


Examples. 


Learn  the 

language  by 

use. 


Its  grammar ; 

Alphabet — 
words,  and 
their  uses. 


§  203.  There  is  but  one  further  idea  to  be 
presented  :  it  is  this, — that  there  are  very  few 
combinations  made  amono;  the  fig;ures,  which 
change,  at  all,  their  signification. 

Selecting  any  two  of  the  figures,  as  3  and  5, 
for  example,  we  see  at  once  that  there  are  but 
three  ways  of  writing  them,  that  will  at  all 
change  their  signification. 

First,  write  them  by  the  side  of  each  )    3  5, 
other ;5  3. 

Second,    write    them,   the    one    ever  i      f, 
the  other )      f- 

Third,  place  a  decimal  point  before  ^      .3, 
each )      .5. 

Now,  each  manner  of  writing  gives  a  differ- 
ent signification  to  both  the  figures.  Use,  how- 
ever, has  established  that  signification,  and  we 
know  it,  as  soon  as  we  have  learned  the  lan- 
guage. 

We  have  thus  explained  what  we  mean  by 
the  arithmetical  language.  Its  grammar  em- 
braces the  names  of  its  elementary  signs,  or 
Alphabet,  —  the  formation  and  number  of  its 
words, — and  the  laws  by  which  figures  are  con- 
nected for  the  purpose  of  expressing  ideas.  We 
feel  that  there  is  simplicity  and  beauty  in  this 
system,  and  hope  it  may  be  useful. 


CHAP.   II.]  ARITHMETIC DEFINITIONS.  187 


NECSSS:T7    of    exact    definitions    and    TERMS. 

§  204.    The   principles    of  every  science   are  Frincipiesoi 
a  collection  of  mental  processes,  having  estab- 
lished connections  with  each  other.     In  every 
branch    of    mathematics,    the    Definitions    and    Dcflnitions 

_  .  'u^d  terms : 

Terms  give  form  to,  and  are  the  signs  of,  cer- 
tain elementary  ideas,  which  are  the  basis  of 
the  science.  Between  any  term  and  the  idea 
which  it  is  employed  to  express,  the  connection 
should  be  so  intimate,  that  the  one  will  always 
suggest  the  other. 

These  definitions  and  terms,  when  their  sig-  when  once 
mtications  are  once  hxed,  must  always  be  used  always  be 
in  the  same  sense.     The  necessity  of  this  is  most 


same  sense. 


urgent.    For,  "in  the  loJioIe  range  of  arithmetical 
science  there  is  no  logical  test  of  ti'ufh,  but  in      Reason. 
a  Cunformity  of  the  reasoning  to  the  definitions 
:nd  terms,  or  to  such  princij)les  as  have  been 
established  from  them." 

§  205.   With  these  principles,  as   guides,  we    Definitions 

^     ,  ,     ^     .    .  ,      and  terms 

propose  to  examine  some  oi  the  dennitions  and    examined, 
terms  which  have,  heretofore,  formed   the  basis 
of  the  arithmetical  science.     We  shall  not  con- 
fine our  quotations  to  a  single  author,  and  shall        ' 
make  only  those  which  fairly  exhibit  the  gen- 
eral use  of  the  terms, 


188  MATHEMATICAL    SCIENCE.  [bOOK  II. 

It  is  said, 
Number  de        "  Numhev  signifies   a  unit,  or   a  collection  of 

fliic-d.  .     .       ,, 

units. 
How  "  The  common  method  of  expressing  numbers 

is  by  the  Arabic  Notation.     The  Arabic  method 
employs  the  following  ten  characters^,  ov figures," 
&c. 
Names  of  the       "The  first  nine  are  called  significant  figures, 

characters.  i  i  i  i  i 

because  each  one  always  has  a  value,  or  denotes 
some  number." 

And  a  little  further  on  we  have, 
Fisures  have       "  The  different  values  which  figures  have,  are 

valma. 

called  simple  and  local  values." 

The  definition   of  Number  is  clear  and  cor- 

Number     rcct.     It  is  a  general   term,  comprehending   al. 

fined:       ^^^  phrascs  which  are   used,   to  express,  either 

separately  or  in  connection,  one  or  more  things 

Also  figures,  of  the  samc  kind.  So,  likewise,  the  definition 
of  figures,  that  they  are  characters,  is  also  right. 

Definition  de-  But  mark  how  soon  these  definitions  are  de- 
parted from.  The  reason  given  why  nine  of  the 
figures  are  called  significant  is,  because  "  each 
one  always  has  a  value,  or  denotes  some  num- 
ber."    This  brings   us  directly  to  the  question. 

Has  a  figure  whether  a  figure  has  a  value;  or,  whether  it  is 
a  mere  representatioe  of  value.  Is  it  a  number 
or  a  character  to  represent    number  ?     Is   it   a 

It  is  merely  .  7     7  -j       t      •       i    r^         i  1  7 

a  character:  quantity  ov  symool  f     It  is  denned  to  be  a  char' 


rilAP.   II.]  ARITHMETIC DEFINITIONS,  189 

acter  which  stands  for,  or  expresses  a  number. 
Has  it  any  other  signification?  How  then  can 
we  say  that  it  has  a  value — and  how  is  it  possi-  iias  novaiur. 
ble  that  it  can  have  a  simple  and  a  local  value  ? 
The  things  which  the  figures  stand  for,  may 
change  their  value,  but  not  the  figures  them- 
selves. Indeed,  it  is  very  di/ficult  for  John  to 
perceive  how  the  figure  2,  standing  in  the  sec-    but  stands 

.  .  for  value. 

ond  place,  is  ten  times  as  great  as  the  same  iig- 

ure  2  standing  in  the  first  place  on  the  right! 

although  he  will  readily  understand,  when   the 

arithmetical  language  is  explained  to  him,  that 

the  UNIT  of  one  of  these  places  is  ten  times  as  unit  of  place. 

great  as  that  of  the  other. 

§  206.  Let  us  now  examine  the  leading  defi-  Leadi-ig  deo 
nition  or  principle  which  forms  the  basis  of  the 
arithmetical  lancruao-e.     It  is  in  these  words : 

"  Numbers   increase  from   right   to   left    in    a   of  number. 
tenfold  ratio  ;  that  is,  each  removal  of  a  figure 
one  place    towards   the  left,  increases    its   value 
ten  times." 

Now,  it  must  be  remembered,   that    number     Does  not 
has   been    defined    as    signifying   "a  unit,   or  a    thedefini- 
collection  of  units."     How,  then,  can  it  have  a   «""  "^^fo^ 
right  hand,  or  a  left  ?  and  how  can  it  increase 
from  right   to    left    in  a   tenfold  ratio  ?"     The 
explanation   given   is,   that   "each  removal  of  a 


190  MATHEMATICAL     SCIENCE.  [bOOK  II. 


Explanation.  figuvB  0716  pluce  toicavds  the  left,  incrcases   its 
value  ten  times." 

Number,  signifying  a  collection  of  units,  must 
Increase  of   iiecessarily   increase    according    to    the    law  by 

numbers  has       i  •    i       i  •  i  •        i  i      i  i 

noconneciion  ■vvhich  thcsc  units  are  combmed  ;  and  that  law 

with  figures.  ^^  increase,   whatever  it  may  be,   has  not  the 

slightest  connection  with  the  figures  which  are 

used  to  express  the  numbers. 

Ratio.  Besides,    is    the    term    ratio    (yet    undefined), 

one  which  expresses  an  elementary  idea?     And 

"Tenfold     is  the  term,  a  "  tenfold  ratio,"  one  of  sufficient 

simplicity  for  the  basis  of  a  system? 

Does,  then,  this  definition,  which  in  substance 

is  used  by  most  authors,  involve   and  carry  to 

Four  leading  the  mind  of  the  young  learner,  the  four  leading 

numberl^    idcas  which  form  the  basis  of  the  arithmetical 

notation  ?  viz.  : 

First.  1st.  That  numbers  are  expressions  for  one  or 

more  things  of  the  same  kind. 
Second.         2d.  That   numbers   are  expressed   by  certain 
characters    called  figures  ;   and  of  which    there 
are  ten. 
Third.  3d.    That    each    figure    always    expresses    as 

many  units  as  its  name  imports,  and  no  more. 
Fourth.  4th.  That  the  Mnd  of  thing  which   a  figure 

expresses  depends  on  the  place  which  the  figure 
occupies,  or  on  the  value  of  the  units,  indicated 
in  some  other  way. 


CHAP.   II.]  ARITHMETIC DEFINITIONS.  191 

Place  is  merely  one  of  the  forms  of  language      Place; 
by  which  we  designate   the  unit  of  a  number,     itaosce. 
expressed  by  a  figure.     The  definition  attributes 
this  property  of  place  both  to  number  and  fig- 
ures, while  it  belongs  to  neither. 


§  207.  Having  considered  the  definitions  and 
terms  in  the  first  division  of  Arithmetic,  viz.  in 
notation  and  numeration,  we  will   now  pass  to  Definitions  in 

,  ,        .         *    1  1  •   •  Addition : 

the  second,  viz.  Addition. 

The  following  are  the  definitions  of  Addition, 
taken  from  three  standard  works  before  me : 

"  The  putting  together  of  two  or  more  num-       First 
bers   (as  in   the   foregoing  examples),  so  as  to 
make  one  whole  number,  is  called  Addition,  and 
the  whole  number  is  called  the  sum,  or  amount."    ' 

"  Addition  is  the  collecting  of   numbers    to-     second, 
gether  to  find  their  sum." 

"  The  process  of  uniting  two   or  more  num-      Third. 
hers  together,  so  as  to  form  one  single  number, 
IS  called  Addition." 

"  The  answer,  or  the  number  thus  found,  is 
called  the  sum,  or  amount." 

Now,  is  there  in  either  of  these   definitions     Defects, 
any    test,   or   means    of  determining  when    the 
pupil  gets  the  thing  he  seeks  for,  viz.  "  the  sum 
of  two  or  more  numbers  ?"     A^o  previous  defi-     Reaaoa 
nition  has  been  given,  in   either  work,  of  the 


192  MATHEMATICAL     SCIENCE.  [liUOKll 

term   sum.     How  is  the  learner  to  know,  what 
he  is  seeking  for,  unless  that  thing  be  defined? 
Noprin-         Suppose  that  John  be  required  to  find  the  sum 

ciple  as  a 

standard.  01  the  numbers  3  and  5,  and  pronounces  it  to 
be  10.  How  will  you  correct  him,  by  showing 
that  he  has  not  conformed  to  the  definitions  and 
rules  ?  You  certainly  cannot,  because  you  have 
established  no  test  of  a  correct  process. 

But,  if  you  have  previously  defined  sum  to  be 
a  number  which  contains  as  many  units  as  there 
are  in  all  the  numbers  added :  or,  if  you  say, 
Correct defl-       "Addition   is   the  process  of  uniting  two  or 
more  numbers,  in  such  a  way,  that  all  the  units 
.    which  they  contain  may  be  expressed  by  a  sin- 
gle number,  called  the  sum,  or  sum  total ;"  you 
*      will  then  have  a  test  for  the  correctness  of  the 
civesatest.  pi'ocess    of   Addition;    viz.    Does    the    number, 
which  you  call  the  sum,  contain  as  many  units 
as  there  are  in  all   the  numbers   added  ?     The 
answer  to  this  question  will  show  that  John  is 
wrong. 

Deiinitions  jf       §  308.    I  wiU    now   quote    the   definitions    of 
Fractions    from    the    same   authors,   and  in   the 
same  order  of  reference. 
Fii-at.  "  We  have  seen,  that  numbers  expressing  whole 

things,  are   called  integers,  or  whole  numbers  ; 
but   that,  in   division,  it   is  often  necessary  to 


CHAP,   II.]  ARITHMETIC EEFINITIONl 


193 


Second. 
Tliiid. 


Tenn  fraction 
defined. 


Ideas 


divid".  or  break  a  whole  thing  into  parts,  and 
that  these  parts  are  called  fractions,  or  broken 
numbers." 

"  Fractions  are  parts  of  an  integer." 

"  When  a  number  or  thing  is  divided  into 
equal  parts,  these  parts  are  called  Fractions." 

Now,  will  either  of  these  definitions  convey 
to  the  mind  of  a  learner,  a  distinct  and  exact 
idea  of  a  fraction  ? 

The  term  "  fraction,"  as  used  in  Arithmetic, 
means  one  or  more  equal  parts  of  something 
regarded  as  a  whole  :  the  parts  to  be  expressed 
in  terms  of  the  thing  divided  considered  as  a 
UNIT.  There  are  three  prominent  ideas  which 
the  mind  must  embrace  : 

1st.  That  the  thing  divided  be  regarded  as  a 
standard,  or  unity  ; 

2d.  That  it  be  divided  into  equal  parts ; 

3d.  That  the  parts  be  expressed  in  terms  of 
the  thing  divided,  regarded  as  a  unit. 

These  ideas  are  referred  to  in  the  latter  part 
of  the  first  definition.  Indeed,  the  definition 
would  suggest  them  to  any  one  acquainted  with 
the  subject,  but  not,  we  think,  to  a  learner. 

In  the  second  definition,  neither  of  them  is 
hinted  at.  Take,  for  example,  the  integer  num- 
ber 12,  and  no  one  would  say  that  any  one  part 
of  this  number,  as  2,  4,  or  6,  is  a  fraction. 

13 


First. 


Second, 
Tliird. 


The  defini- 
tions exam 
ined: 


Is  a  frac- 
tion part  of 
an  integer 


194  MATHEMATICAL     SCIENCE.  [bOOK  11 

Third  The  third  definition  would  be  perfectly  accu- 

definition ;  .  .  r  i  i  i  •         m       i 

rate,  by  inserting  alter  the  word  "  thing,  the 
words,  "  regarded  as  a  whole."  It  very  clearly 
expresses  the  idea  of  equal  parts,  but  does  not 
In  what  de-  present  the  idea  strongly  enough,  that  the  thing 
divided  must  be  regarded  as  unity,  and  that  the 
parts  must  be  expressed  in  terms  of  this  unity. 

§  209.  I  have  thus  given  a  few  examples,  illus- 

Necessity  of  ti'atiug  the  necessity  of  accurate  definitions  and 
terms.  Nothing  further  need  be  added,  except 
the  remark,  that  terms  should  always  be  used  in 
the  same  sense,  precisely,  in  which  they  are  de- 
fined. 
Objection         To  some,  perhaps,  these  distinctions  may  ap- 

of  thought    P6^r  over-nice,   and  matters   of  little    moment. 

nnd language,  j^  ^^^  ^^  supposcd  that  a  general  impression, 
imparted  by  a  language  reasonably  accurate, 
will  suffice  very  well ;  and  that  it  is  hardly 
worth  while  to  pause  and  weigh  words  on  a 
nicely-adjusted  balance. 

Any  such  notions,  permit  me  to  say,  will  lead 
to  fatal  errors  in  education. 

Definitions  in       It  is  iu  mathematical  science  alone  that  words 

''"™'^'  ■  are  the  signs  of  exact  and  clearly-defined  ideas. 

It  is  here  only  that  we  can  see,  as  it  were,  the 

very  thoughts  through  the  transparent  words  by 

which  they  are  expressed.     If  the  words  of  the 


CHAP.   II.]  ARITHMETIC SUBJECTS.  196 


reason  cur 
redly. 


definitions  are  not  such  as  convey  to  the  mind     Must  be 

GXflCt  to 

of  the  learner,  the  fundamental  ideas  of  the 
science,  he  cannot  reason  upon  these  ideas ; 
for,  he  does  not  apprehend  them ;  and  the  great 
reasoning  faculty,  by  which  all  the  subsequent 
principles  of  mathematics  are  developed,  is  en- 
tirely unexercised.* 

It   is   not  possible   to  cultivate   the    habit    of  camuii other- 

....  .   ,  .  wise  cultivate 

accurate   thmking,  without  the  aid   and  use  of     habits  of 
exact  language.     No  mental  habit  is  more  use- 
ful than  that  of  tracing  out  the  connection  be- 
tween ideas  and  language.     In  Arithmetic,  that 
connection    can    be    made    strikingly    apparent.    Connection 
Clear,    distinct    ideas — diamond    thoughts — may    worJsand 
be  strung  through   the   mind   on  the  thread  of  ""'^^^'^'*.« 

~  o  anthinetic. 

science,  and  each  have  its  word  or  phrase  by 
which  it  can  be  transferred  to  the  minds  of 
others. 


now    SHOULD    THE    SUBJECTS    BE    PRESENTED  ? 

§  210.  Having  considered  the  natural  connec-       vvhai 
lion   of  the   subjects   of  arithmetic   with   each   considmi<t 
other,  as  branches  of  a  single  science,  based  on 
a  single  unit ;    and    having    also    explained    the 
necessity   of  a   pei-spicuous    and    accurate  Ian- 


*  Section  200. 


196  MATHEMATICAL     SCIENCE.  [rOOK   II 

How  ought   guage ,  we  come  now  to  that  important  inquiry, 

the  subjects 

tobepie-     How  ought  those  subjects  to  be  presented  to  the 

mind  of  a  learner?    Before  answering  this  ques- 

Two objects  tioH,  wc  should  reflect,  that  two  important  ob- 

in  studying     .  i         i  .   i  i  r  •         i  i  r        •    \ 

arithmetic:   JGcts  should  be  sought  alter  in  the  study  oi  arith- 
metic : 
First  1st.    To   train    the    mind    to  habits    of  clear, 

quicjv,    and    accurate    thought — to    teach    it    to 
apprehend  distinctly — to  discriminate  closely — 
to  judge  truly — and  to  reason  correctly  ;  and. 
Second.  2d.    To    give,    in    abundance,    that   practical 

knowledge   of  the   use   of  figures,   in   their  va- 
rious applications,  which  shall  illustrate  the  stri- 

Artofarith    king  fact,  that  the  art  of  arithmetic  is  the  most 

metic. 

important   art   of  civilized  life — being,   in  fact, 
th  ■  foundation  of  nearly  all  the  others. 

How  first  im        §211.   It  is  Certainly  true,  that   most,  if  not 

presslons  art  .  i        i  i 

made.       -il-  the  elementary  notions,  whether  abstract  or 
pi-actical — that    is,  whether    they  relate    to    the 
science    or    to    the   art  of  arithmetic,   must   be 
made  on  the  mind  by  means  of  sensible  objects. 
Because  of  this  fact,  many  have  supposed  that 
Is  reason-     the  processes   of  reasoning   are   all   to   be   con- 
ducted by    ducted  by  the  same  sensible  objects;    and  that 
sensible     q^qyv  abstract  principle  of  science  is  to  be  de- 

objects?  •'  I  r 

veloped    and    established    ly    means    of    sofas, 
chairs,  apples,  and  horses.     There  seems  to  be 


CHAP.   II.]  ARITHMETIC SUBJECTS.  197 

an   impression    that    because    blocks    are    useful     sensible 

...  ,  .  ,  111  1  ;  /•  objects  useful 

aids    in    teaching    the    alphabet,    that,    tlierefore  m  acquiring 
they    can    be    used    advantageously    in    reading  ^'^^  simplest 

•'  o  J  o      elements : 

Milton  and  Shakspeare.  This  error  is  akin  to 
that  of  attempting  to  teach  practically.  Geog- 
raphy and  Surveying  in  connection  with  Geom-       Enor    ■ 

1  II-  I  ^  r  1  1         of  carrviug 

etry,  by  calling  the  angles  oi  a  rectangle,  north,  t^e.^ 

south,  east,  and  west,  instead  of  simply  designa-  ^'^y°^'^- 
ting  them  by  the  letters  A,  B,  C,  and  D. 

This  false  idea,  that  every  principle   of  sci-  False  idea: 
ence    must    be    learned  practically,    instead    of 

being  rendered  practical  hy  its  applications,  has  its  effects. 
been  highly  detrimental  both  to  science  and  art. 

A  mechanic,  for  example,  knowing  the  height  Example 

of  his  roof  and  the  width  of  his  building,  wishes  catiun  of 

to  cut  his  rafters  to  the  proper  length.     If  he  ^";''^®"^'^' 

r      I  o  pniiciple: 

calls  to  his  aid  the  established,  though  abstract 
principles  of  science,  he  finds  the  length  of  his 
rafter,  by  the  well-known  relation  between  the 
hypothenuse  and  the  two  sides  of  a  right-angled 
triangle.  If,  however,  he  will  learn  nothing  ex- 
cept  practically,  he  must  raise  his  rafter  to  the    of  learning 

practically. 

roof,  measure  it,  and  if  it  be  too  long  cut  it  ofi, 
if  too  short,  splice  it.  This  is  the  practical  way 
of  learning  things. 

The  truly  practical  way,  is  that  in  which  skill       "^""^ 

practical. 

is  guided  by  science. 


Do  the  principles  above  stated  find  any  appl 


198  MATHEMATICAL     SCIENCE.  [bOOK  H. 


cation  in  considering  the  question.  How  should 
cau        ar'thmetic    be   taught?     Certainly  they  do.     If 
arithmetic    be    both    a    science    and    an    art,   it 
should  be  so  taught  and  so  learned. 


I'ri.icipies        §  212.  The  principles  of  every  science  are  gen- 
eral and  abstract  truths.     They  are  mere  ideas. 
What      primarily  acquired  through  the  senses  by  experi- 

theyare:  t        i    i  r        n         • 

ence,  and  generalized  by  processes  oi  reilection 
Wise       and  reasoning;  and  when  understood,  are  certain 

lo  use  them.  ...  i  •    i       i  i  ■       i  i 

guides  m  every  case  to  which  they  are  applicable. 
■  If  we  choose  to  do  without  them,  we  may.     But 
is  it  wise  to  turn  our  heads  from  the  guide-boards 
and  explore  every  road  that  opens  before  us  ? 

Now,  in  the  study  of  arithmetic  those  princi- 
ples of  science,  applicable  to  classes    of  cases, 
UHien      should  always  be  taught  at  the  earliest  possible 
uiey  should   moment.      The   mind    should    never   be    forced 
'^"'°  '    through  a  long  series  of  examples,  without  ex- 
The  methods  plauatiou.     Oiic  or  two  cxamplcs  should  always 

pointed  out.  r  ^  •        •    i 

precede  the  statement  oi  an  abstract  principle, 
or  the  laying  down  of  a  rule,  so  as  to  make  the 
anguage  of  the  principle  or  rule  intelligible. 
But  to  carry  the  learner  forward  through  a 
Principles  scrics  of  them,  before  the  principle  on  which 
ce-u  they  depend  has  been  examined  and  stated,  is 
forcing  the  mind  to  advance  mechanically — it 
is  lifting  up  the  rafter  to  measure  it,  when  its 


CHAP.  II.]  ARITHMETIC TEXT-BOOKS.  199 

exact  length   could  be  easily  determined  by  a 
rule  of  science. 

As  most  of  the  instructian  in  arithmetic  must      Books: 
be  given  with  the  aid  of  books,  we  feel  unable 
to  do  justice  to  this  branch  of  the  subject  with-     Necessity 

...  r  1  ■  1  for  treating 

out  submittmg  a  lew  observations  on  the  nature     of  them. 
of  text-books  and  the  objects  which  they  arc  in- 
tended to  answer. 


TEXT -BOOKS. 

§  213.   A  text-book  should  be  an  aid  to  the   Text-book 
teacher   in    imparting   instruction,    and    to    the 
learner  in  acquiring  knowledge. 

It  should  present   the  subjects  of  knowledge     What  it 
in  their  proper  order,  with  the  branches  of  each 
subject  classified,  and  the  parts  rightly  arranged. 
No  text-book,  on  a  subject  of  general  knowledge.     Selection 

11     1  •       1  r     I  1  •  "'^  subjects 

can  contain  all  that  is  known  oi  the  subject  on    necessary. 

which   it   treats ;  and  ordinarily,  it  can   contain 

but  a  very  small  part.     Hence,  the  subjects  to 

be  presented,  and  the  extent  to  which  they  are    Difficuiues 

,  ,  „.,....  of  selection 

to  be  treated,  are  matters  oi  nice  discrimination 
and  judgment,  about  which  there  must  always 
be  a  diversity  of  opinion. 

§  214.  The  subjects  selected  should  be  leading    subjects: 
ones,  and  those  best   calculated   to  unfold,  ex- 


200 


M.\THEMAriCAL     SCIENCE, 


[book  II. 


-    plain,  and  illustrate  the  principles  of  the  science. 
How       They  should  be  so  presented  as  to  lead  the  mind 

presented. 

to  analyze,  discriminate,  and  classify ;  to  see 
each  principle  separately,  each  in  its  combina- 
tion with  others,  and  all,  as  forming  an  harmo- 
nious whole.  Too  much  care  cannot  be  be- 
suggestive    stowed    in    forming    the     suggestive    method    of 

method:  .  . 

arrangement :    that    is,   to  place    the   ideas    and 
principles   in  such   a  connection,  that  each  step 
Reason  for.    shall  prepare  the  mind  of  the  learner  for  the  next 
in  order. 


Object  §  215.   A  text-book  should  be  constructed  for 

of  a  textr  .    „        .    .  . 

book:  the  purpose  ol  furnishing  the  learner  wdth  the 
keys  of  knowledge.     It  should  point  out,  explain, 

Nature;  and  illustrate  by  examples,  the  methods  of  in- 
vestigating and  examining  subjects,  but  should 
leave  the  mind  of  the  learner  free  from  the  re- 
straints of  minute  detail.  To  fill  a  book  with 
the  analysis  of  simple  questions,  which  any  child 
can  solve  in  his  own  way,  is  to  constrain  and 
force  the  mind  at  the  very  point  where  it  is  ca- 
pable of  self-action.  To  do  that  for  a  pupil, 
which  he  can  do  for  himself,  is  most  unwise. 


Useless 
detail; 


Should  §  216.    A    text-book  on   a  subject  of  science 

not  be  his-  ii--iAr' 

toiicai.      should  not  be  historical.     At  first,  the  minds  of 
children  are  averse  to  whatever  is  abstract,  be- 


CUAP.   II.]  ARITHMETIC TEXT-BOOKS.  201 

cause  what  is  abstract  demands  thought,  and  Reasons, 
thinking  is  mental  labor  from  which  untrained 
minds  turn  away.  If  the  thread  of  science  be 
broken  by  the  presentation  of  facts,  having  no 
connection  with  the  argument,  the  mind  will 
leave  the  more  rugged  path  of  the  reasoning, 
and  employ  itself  with  what  requires  less  effort 
and  labor. 

The  optician,  in  his  delicate  experiments,  ex-   illustration. 
eludes  all  light  except  the  beam  which  he  uses : 
so,    the    skilful    teacher    excludes    all    thoughts 
excepting   those  which  he    is   most  anxious  to 
impress. 

As  a  general  rule,  subject  of  course  to  some 
exceptions,  but  one  method  for  each  process  One  methcd. 
should  be  given.  The  minds  of  learners  should 
not  be  confused.  If  several  methods  are  given,  Reasons 
it  becomes  difficult  to  distinguish  the  reasonings 
applicable  to  each,  and  it  requires  much  knowl- 
edge of  a  subject  to  compare  different  methods 
with  each  other. 

§  217.  It  seems  to  be  a  settled  opinion,  both     kow  the 
among  authors  and  teachers,  that  the  subject  of     divided, 
arithmetic  can  be  best  presented  by  means  of 
three  separate  works.     For  the  sake  of  distinc- 
tion, we  will  designate  them  the  First,  Second, 
and  Third  Arithmetics. 


202  MATHEMATICAL     SCIENCE.  [  BOOK    11. 

We  will  now  explain  what  we  suppose  to  be 
the  proper  construction  of  each  book,  and  the 
object  for  which  each  should  be  desijrned. 

FIRST     ARITHMETIC. 

First  §  218.    This   book   should  give    to    the    mind 

Arithmetic: 

its   first  direction  in   mathematical  science,  and 

its    first    impulse    in    intellectual    development. 

Its        Hence,  it   is   the   most    important   book  of  the 

importance. 

series.     Here,  the  faculties  of  apprehension,  dis- 
crimination, abstraction,  classification  and  com- 
parison, are   brought  first   into  activity.     Now, 
How       to  cultivate  and  develop  these  faculties  rightly, 

the  subjects 

must  be     wc   must,   at   first,   present  every   new  idea   by 
v^eaeae  .    j^g^j^g   q|-  ^  sensible   objcct,  and   then   immedi- 
ately drop  the  object  and  pass  to  the  abstract 
thought. 
Order  Wc  must  also  prcscut  the  ideas  consecutively; 

of  the  ideas. 

that  is,  in  their  proper  order ;  and  by  the  mere 
method  of  presentation  awaken  the  comparative 
and  reasoning  faculties.  Hence,  every  lesson 
should  contain  a  given  number  of  ideas.  The 
Construction  idcas   of  cach  Icssoii,  beginning  with   the  first, 

of  the  lessons.     ,        ,  ,         ,  .  ,  ,      .  ,      , 

should  advance  in  regular  gradation,  and  the 
lessons  themselves  should  be  regular  steps  in 
the  progress  and  development  of  the  arithmeti- 
cal science. 


CHAP.   II.]  ARITHMETIC TEXT- BO  OK  3. 


303 


§  219.  The  first  lesson  should  merely  contain 
representations  of  sensible  objects,  placed  oppo- 
site names  of  numbers,  to  give  the  impression 
of  the  meanings  of  these  names :  thus, 


One- 
Two 
Three 
&c. 


*  *  * 
&c. 


And  with  young  pupils,  more  striking  objects 
should  be  substituted  for  the  stars. 

In  the  second  lesson,  the  words  should  be  re- 
placed by  the  figures  :  thus, 


1  - 

2  - 

3  - 

&c. 


*  tK-  * 
&c. 


In  the  third  lesson,  I  would  combine  the  ideas 
of  the  first  two,  by  placing  the  words  and  fig- 
ures opposite  each  other  :  thus. 


First 
Icssou. 


What  it 

should  con- 

taiu. 


The  Roman  method  of  representing  numbers 
should  next  be  taught,  making  the  fourth  lesson  • 
viz., 


Second 
lessou. 


One    -     - 

-     -     1 

Four  -     - 

-     -     4 

Two  -     - 

-     -     2 

Five  -     - 

-     -     5 

Third 

Three     - 

-     -     3 

Six     -     = 

=     -     6 

lessoa 

&c. 

(fee. 

&c. 

&c. 

204 


MATHEMATICAL     SCiENCE. 


[bCOK  11. 


Fourth 

lesson. 


Roman 
method. 


One  - 
Two  - 
Three 
&c. 


I. 

Four 

II. 

Five 

III. 

Six 

&c. 

&c. 

IV. 
V. 

VJ. 
&c. 


First 

ten  combi- 

Dations : 


§  220.  We  come  now  to  the  first  ten  com- 
binations of  numbers,  which  should  be  given  in 
a  separate  lesson.  In  teaching  them,  we  must, 
of  course,  have  the  aid  of  sensible  objects.  We 
teach  them  thus : 


One 

and 

one 

are 

how 

many  ? 

How 

taught  by 
thiugs: 

*• 
One 

and 

* 
two 

are 

how 

many  ? 

One 

and 

three 

are 

how 

many  ? 

* 

*  *■  * 

&c. 

&c. 

&c., 

through  all  the  combinations :  after  which,  we 
How  in  pass  to  the  abstract  combinations,  and  ask,  one 
and  one  are  how  many  ?  one  and  two,  how 
many  ?  one  and  three,  &c. ;  after  which  we 
express  the  results  in  figures. 

We  would  then  teach  in  the  same  manner,  in 
a  separate  lesson,  the  second  ten  combinations ; 
then  the  third,  fourth,  fifth,  sixth,  seventh,  eighth, 
ninth,  and  tenth.  In  the  teaching  of  these  com- 
Words  used,  binations,  only  the  words  from  one  to  twenty 
will  have  been  used.     We  must  then  teach  the 


Second 
ten  combina- 
tions. 


ARITHMETIC TEXT-BOOKS. 


205 


combinations  of  which  the  results  are  expressed     Further 

11  1      r  111  combina- 

by  the  words  irora  twenty  to  one  hundred.  tions. 


How 
they  appear, 


§  221.  Having  done  this,  in  the  way  indi-  Results. 
cated,  the  learner  sees  at  a  glance,  the  basis  on 
which  the  system  of  common  numbers  is  con- 
structed. He  distinguishes  readily,  the  unit  one 
from  the  unit  ten,  apprehends  clearly  how  the 
second  is  derived  from  the  first,  and  by  com- 
paring them  together,  comprehends  their  mutual 
relation. 

Having  sufficiently  impressed  on  the  mind  ot 
the  learner,  the  important  fact,  that  numbers  are 
but  expressions  for  one  or  more  things  of  the 
same  kind,  the  unit  mark  may  be  omitted  in  the     Unit  mark 

,  .         .  1  ■    1     f  11  omitted. 

combmations  which  lollow. 


Same 
method  in 
the  other 

rules. 


§  222.  With  the  single  difference  of  the  omis- 
sion of  the  unit  mark,  the  very  same  method 
should  be  used  in  teaching  the  one  hundred 
combinations  in  subtraction,  the  one  hundred 
and  forty-four  in  multiplication,  and  the  one 
hundred  and  forty-four  in  division. 

When  the  elementary  combinations  of  the  four 
ground  rules  are  thus  taught,  the  learner  looks    Results  of 

,        ,        ,  ,  .  f.  I  .  .        the  method 

back  through  a  series  oi  regular  progression,  m 
which  every  lesson  forms  an  advancing  step, 
and  where  all  the  ideas  of  each  lesson   have  a 


206  MATHEMATICAL     SCIENCE.  [bOOK   II. 

mutual  and  intimate  connection  with  each  other. 

Are  they     Will  uot  such  a  systcm   of  teaching  train  the 

mind  to  the  habit  of  regarding  each  idea  sepa- 

Thc        rately — of  tracing  the  connection  between  each 

give.  "    new  idea  and  those  previously  acquired — and  of 

comparing  thoughts  with  each  other? — and  are 

not  these  among  the  great  ends  to  be  attained, 

by  instruction  ? 

§  223.  It  has  seemed  to  me  of  great  import- 
Figures      ance  to  use  figures  in  the  very  first  exercises  of 
useTeariy.    arithmetic.      Unless  this  be  done,  the  operations 
must  all  be  conducted  by  means  of  sounds,  and 
Reasons,     the  pupll  is  thus  taught  to  regard  sounds  as  the 
proper   symbols    of   the    arithmetical    language, 
conse-      This    habit  of  mind,  once  firmly  fixed,  cannot 
asing  words  be  easily  eradicated ;  and  when  the  figures  are 
°"'^        learned   afterwards,   they  will  not    be    regarded 
as    the    representatives    of   as    many    things    as 
their  names  respectively  import,  but  as  the  rep- 
resentatives   merely    of   familiar    sounds   which 
have  been  before  learned. 

This  would  seem  to  account  for  the  fact, 
about  which,  I  believe,  there  is  no  difference  of 
Oral  opinion ;  that  a  course  of  oral  arithmetic,  ex- 
tending over  the  whole  subject,  without  the  aid 
and  use  of  figures,  is  but  a  poor  preparation 
for  operations  on  the  Sxdte.     It  may,  it  is  true^ 


CHA  '.   n.]  ARITHMETIC TEXT-BOOKS.  207 

sharpen  and   strengthen   the   mind,   and  give   it      what 

11  1         1         •       •       •      1       1  ''  '"''y  **°- 

development :   but  does  it  give  it  that  language 

and  those  habits  of  thought,  which  turn  it  into     what  it 

the  pathways  of  science  ?     The  language   of  a 

science    affords    the    tools    by  which    the    mind    Language 

ofaiithmetic: 

pries  into  its  mysteries   and  digs  up  its  hidden 
treasures.     The  language  of  arithmetic  is  formed 
from  the  ten  figures.     By  the   aid  of  this   Ian-     its  uses, 
guage   we   measure   the   diameter  of   a  spider's 
web,    or    the    distance    to    the    remotest   planet       what 
which  circles  the  heavens  ;    by  its  aid,  we  cal- 
culate the  size  of  a  grain  of  sand  and  the  mag- 
nitude of  the  sun  himself:  should  we  then  aban- 
don a  language  so  potent,  and  attempt  to  teach     its  value, 
arithmetic   in    one   which   is    unknown    in    the 
higher  departments  of  the  science  ? 

§  224.  We  next  come  to  the  question,  how    Fractions 
the  subject  of  fractions  should  be  presented  in 
an  elementary  work. 

The  simplest  idea  of  a  fraction   comes  from     simplest 
dividing  the  unit  one  into  two  equal  parts.     To 
ascertain  if  this  idea  is  clearly  apprehended,  put       how 

,  .  ^_  ,  impressed 

the   question.   How  many   halves    are    there    m 

one  ?     The  next  question,  and  it  is  an  import-       Next 

,  .         TT  II  1  •  question. 

ant  one,  is  this  :  How  many  halves  are  there  in 
one  and  one-half?  The  next,  How  many  halves 
in   two?     How  many  in   two  and  a  half?     In 


208  MATHEMATICAL     SCIENCE,  [bOOKIJ. 

three  ?     Three  and  a  half?  and  so  on  to  twelve. 
Rosuits.     You   will   thus   evolve   all   the  halves  from   the 
units  of  the   numbers  from   one  to   twelve,  in- 
clusive.    We   stop  here,   because    the    multipli- 
cation table  goes  no  further.     These   combina- 
First  lesson,  tious  should  be  embraced  in  the  first  lesson  on 
fractions.      That  lesson,  therefore,  will  teach  the 
it.s  extent,    relation  between  the  unit  1  and  the  halves^  and 
^oint  out  how  the  latter  are  obtained  from  the 
former. 

Second  §  225.   The  second  lesson  should  be  the  first, 

reversed.  The  first  question  is,  how  man}- 
Grades  wholc  things  are  there  in  two  halves  ?  Sec- 
ond, How  many  whole  things  in  four  halves  ? 
How  many  in  eight  ?  and  so  on  to  twenty-four 
halves,  when  we  reach  the  extent  of  the  division 
E.-cteniQf  table.  In  this  lesson  you  will  have  taught  the 
pupil  to  pass  back  from  the  fractions  to  the  unit 
from  which  they  are  derived. 

Kiindamentai       §  22G.  You  havc  thus  taught  the  two  funda- 
prinLipLs.    j^gj^^|.^|  principles  of  all   the  operations  in  frac- 
tions :  viz. 
Fii-st.  1st.  To  deduce  the  fractional  units  from  in- 

tegral units ;  and, 
Becond.         2dly.  To  dcducc  integral  units  from  fractional, 
units 


CHAP.   II.]  ARITH.VIETIC TEXT-BOOKS. 


209 


§  227.  The  next  lesson  should  explain  the  law     Lessons 

explaining 

i)y  which  the  thirds  are  derived  from  the  units      thirds, 
from  1  to  12  inclusive ;  and  the  following  lesson 
the  manner  of  changing  the  thirds  into  integral 
units. 

The  next  two  lessons  should  exhibit  the  same     Fourths 

and  other 

operations    performed    on    the   fourth,    the   next     fractions. 
two   on    the   fifth,    and   so   on    to    include    the 
twelfth. 


§  228.  This  method  of  treating  the  subject  of  Advantages 

of  the 

fractions  has  many  advantages  :  method, 

1st.  It  points  out,  most  distinctly,  the  relations 
between  the  unit  1  and  the  fractions  which  are       First 
derived  from  it. 

2d.  It  points  out  clearly  the  methods  of  pass-      second, 
ing  from  the  fractional  to  the  integral  units. 

3d.    It  teaches  the  pupil  to  handle  and  com-       Third, 
bine  the  fractional  units,  as  entire  things. 

4th.  It  reviews  the  pupil,  thoroughly,  through      Fourth, 
the  multiplication  and  division  tables. 

5th.    It  awakens   and  stimulates   ihe  faculties       Finh. 
of  apprehension,  comparison,  and  classification. 


§  229.    Besides    the    subjects    already  named,  what 

the    First    Arithmetic    should    also    contain    the  Arithmetic 

tables   of  denominate    numbers,  and   collections  ®'^°"'.'^  ""*" 

'  tain. 

of  simple  examples,  to  be  worked  on  the  slate, 

14 


310 


MATHEMATICAL     SCIENCE. 


[book  H 


Examples,    Under  the   direction  of  the   teacher.     It  is   not 

how  taught. 

supposed  that  the  mind  of  the  pupil  is  suffi- 
ciently matured  at  this  stage  of  his  progress  to 
understand  and  work  by  rules. 


What 
should  be 
taught  in 
the  First 
Arithmetic. 


Second. 

Third. 
Fourth. 
Fifth. 


§  230.  In  the  First  Arithmetic,  therefore, 
the  pupil  should  be  taught, 

1st.  The  language  of  figures ; 

2d.  The  four  hundred  and  eighty-eight  ele- 
mentary combinations,  and  the  words  by  which 
they  are  expressed  ; 

3d.  The  main  principles  of  Fractions  ; 

4th.  The  tables  of  Denominate  Numbers;  and, 

5th.  To  perform,  upon  the  slate,  the  element- 
ary operations  in  the  four  ground  rules. 


SECOND     ARITHMETIC. 


Second 
Arithmetic. 


What  it 
should  be. 


§  231.  This  arithmetic  occupies  a  large  space 
in  the  school  education  of  the  country.  Many 
study  it,  who  study  no  other.  It  should,  there- 
fore, be  complete  in  itself  It  should  also  be 
eminently  practical ;  but  it  cannot  be  made  so 
either  by  giving  it  the  name,  or  by  multiplying 
the  examples.    ' 


vracticai         §  232.    The  truly  practical  cannot  be  the  ante- 

npplication  of 

principle.    Cedent,  but  must  be  the  consequent  of  science. 


n.J 


ARITHMETIC TEXT-B(»OKS. 


211 


Hence,    that    general    arrangement    of   subjects  Arranseraeni 
demanded    by    science,    and    already   explained, 
must  be  rigorously  followed. 

But  in   the   treatment  of  the  subjects   them-   Reawinsfor 
selves,  we  are  obliged,  on  account  of  the  limited 
information  of  the  learners,  to  adopt  methods  of 
teaching  less  general  than  we  could  desire. 


§  233.  We  must  here,  again,  begin  with  the 
unit  one,  and  explain  the  general  formation  of 
the  arithmetical  language,  and  must  also  ad- 
here rigidly  to  the  method  of  introducing  new 
principles  or  rules  by  means  of  sensible  objects. 
This  is  most  easily  and  successfully  done  either 
by  an  example  or  question,  so  constructed  as  to 
show  the  application  of  the  principle  or  rule. 
Such  questions  or  examples  being  used  merely 
tor  the  purpose  of  illustration,  one  or  two  will 
answer  the  purpose  much  better  than  twenty  : 
for,  if  a  large  number  be  employed,  they  are  Keasons. 
regarded  as  examples  for  practice,  and  are  lost 
sight  of  as  illustrations.  Besides,  it  confuses 
the  mind  to  drag  it  through  a  long  series  of 
examples,  before  explaining  the  principles  by 
which  they  are  solved.     One  example,  wrought  one  example 

uuder  a  rule. 

under  a  principle  or  rule  clearly  apprehended, 
conveys  to  the  mind  more  practical  informa- 
tion, than  a  dozen  wrought  out  as  independent 


How 
carried  out 


Few 
examples. 


312 


MATHEMATICAL     SCIENCE, 


[book  II. 


Principle,    exevcises.     Let  the  principle  precede  the  prac- 
Practice.     ticc,   ill    all    cascs,   as    soon   as   the  information 

acquired  will   permit.     This   is   the   golden  rule 

both  of  art  and  morals. 


Subjects  §  234.    The    Second   Arithmetic    should   em- 

cmbraced.  n      i  i  •  r  n       • 

brace  all  the  subjects  necessary  to  a  lull  view 
of  the  science  of  numbers  ;  and  should  contain 
an  abundance  of  examples  to  illustrate  their 
Reading:  practical  applications.  The  reading  of  numbers, 
so  much  (though  not  too  much)  dwelt  upon,  is 
an  invaluable  aid  in  all  practical  operations. 

By  its  aid,  in  addition,  the  eye  runs  up  the 
columns  and  collects,  in  a  moment,  the  sum  of 
Bubtraction :  all  the  numbers.  In  subtraction,  it  glances  at 
the  figures,  and  the  result  is  immediately  sug- 
gested. In  multiplication,  also,  the  sight  of  the 
figures  brings  to  mind  the  result,  and  it  is 
reached  and  expressed  by  one  word  instead  of 
five.  In  short  division,  likewise,  there  is  a  cor- 
responding saving  of  time  by  reading  the  results 
of  the  operations  instead  of  spelling  them.  The 
method  of  reading  should,  therefore,  be  con- 
stantly practised,  and  none  other  allowed. 


Its  value 
in  Addition : 


Multi- 
plication : 


Division. 


CHAP.   II.]  ARITHMETIC TEXT-BOOKS.  213 


THIRD     ARITHMETIC. 

§  235.  We  have  now  reached  the  place  where      Thw 

...  1  ,  .  —,,         Arithmetic 

arithmetic    may   be   taught   as   a  science.      Ihe 
pupil,  before  entering  on  the  subject  as  treated   Preparation 
here,  should  be  able  to  perform,  at  least  mechan- 
ically, the  operations  of  the  five  ground  rules. 

Arithmetic  is  now  to  be  looked  at   from  an 
entirely   diflerent    point   of   view.      The   great   view  of  it. 
principles  of  generalization  are  now  to  be  ex- 
plained and  applied. 

Primarily,    the    general    language    of    figures      what 

1  1  II  -I  •  f  I  '*  taught 

must  be  taught,  and  the  striking  fact  must  then    primaiiiy. 

be  explained,  that  the  construction  of  all  integer     * 

numbers    involves    but    a   single    principle,    viz. 

the  law  of  change  in  passing  from  07ie  unit  to  ctueraiiaw 

another.     The  basis  of  all  subsequent  operations 

will  thus  have  been  laid. 

8  236.  Takino;  advantage  of  this  general  law 
which  controls  the    formation    of  numbers,  we     Controls 

.  PI-  1  formation  0/ 

bring  all  the  operations  oi   reduction  under  one    numbers 
single  principle,  viz.  this  law  of  change  in  the 
unities. 

Passing  to  addition,  we  are  equally  surprised     itavahie 
and  delighted   to  find   the   same   principle   con- 
trolling all  its  operations,  and  that  integer  num- 
bers of  all  kinds,  whether  simple  or  denominate, 
may  be  added  under  a  single  rule. 


214  MATHEMATICAL     SCIENCE.  [bOOK   II. 

Advantages       This  view  opens  to  the  mind  of  the  pupil  a 

of  knowing  a        •  i        /^    i  i        r      i  i  t       •         i  c- 

general  Jaw.  Wide  held  ot    thought.     It   IS   the    hrst  illustra- 
tion of  the  great  advantage  which  arises  from 
looking  into   the    laws    by   which    numbers    are 
Subtraction.  Constructed.      In    subtraction,    also,    the    same 
principle  finds  a  similar  application,  and  a  sim- 
ple rule   containing  but   a  few   words  is  found 
applicable  to  all  the  classes  of  integral  numbers. 
In  multiplication  and  division,  the  same  stri- 
king  results    flow    from   the    same    cause ;    and 
General     thus  this  simple  principle,  viz.  the  law  of  change 
bers:       in  passing  fvom  one  unit  of  value  to  another,  is 
the  key  to  all  the  operations  in  the  four  ground 
rules,  whether  performed  on  simple  or  denomi- 
nate numbers.     Thus,  all  the  elementary  opera- 
controis     tions  of  arithmetic   are  linked  to  a  single  prin- 
tion.       ciple,  and  that  one   a  mere  principle  of  arith- 
metical language.     Who  can   calculate   the  la- 
bor,   intellectual    and    mechanical,    which    may 
be  saved   by  a  right   application  of  this  lumin- 
ous principle  ? 

Design  §  237.    It   should   be  the  design   of  a  higher 

arithmetic:  arithmetic  to  expand  the  mind  of  the  learner 
over  the  whole  science  of  numbers ;  to  illus- 
trate the  most  important  applications,  and  to 
make  manifest  the  connection  between  the  sci- 
ence and  the  art. 


CHAP.   II.j  ARITHMETIC TEXT-BOOKS.  315 


It  will  not  answer  these  objects  if  the  methods         its 

requisites. 

of  treating  the  subject  are  the  same  as  in  the 
elementary  works,  where  science  has  to  com- 
promise with  a  want  of  intelligence.  An  ele- 
mentary  is   not  made  a   higher    arithmetic,   by    Must  have 

.    .  .  ...  a  distinctive 

merely  transferring  its  definitions,  its  principles,    ciiaracter. 
and    its   rules   into   a  larger  book,  in   the   same 
order  and  connection,  and  arranging  under  them 
an  apparently  new  set  of  examples,  though  in  fact 
constructed  on  precisely  the  same  principles. 

§  238.    In  the  four  ground  rules,  particularly     construc- 
tion of  exam 
(where,  in  the  elementary  works,  simple  exam-    pies  in  the 

,  •  1        1  •  1  1  four  ground 

pies    must    necessarily    be    given,   because    here       ^^^^ 
they  are  used  both  for  illustration  and  practice), 
the  examples  should  take  a  wide  range,  and  be 
so  selected  and  combined  as  to  show  their  com- 
mon dependence  on  the  same  principle. 

§  239.  It  being  the  leading  design  of  a  series      Design 

.       of  a  Berie& 

of  arithmetics  to  explain  and  illustrate  the  sci- 
ence and  art  of  numbers,  great  care  should  be 
taken  to  treat  all  the  subjects,  as  far  as  their 
different  natures  will  permit,  according  to  the 
same  general  methods.  In  passing  from  one 
book  to  another,  every  subject  which  has  been  subjectf* 
fully  and  satisfactorily  treated  in  the  one,  should  fcredwheu 
be  transferred  to  the  other  with  the  fewest  pos-    ""y'^®'''^ 


216  MATHEfllATICAL     SCIENCE.  [iJIOKlI. 

How  com-    sible  alterations;  so  that  a  pupil  shall  not  have 

mon  subjects  i  ■    i      ,         , 

maybe  to  learn  under  a  new  dress  that  which  he  has 
already  fully  acquired.  They  who  have  studied 
the  elementary  work  should,  in  the  higher  one, 
either  omit  the  common  subjects  or  pass  them 
over  rapidly  in  review. 

The  more  enlarged  and  comprehensive  views 

Eeafions.     which  should  be  given  in  the  higher  work  will 

thus  be  acquired  with  the  least  possible  labor,  and 

the  connection  of  the  series  clearly  pointed  out. 

This  use  of  those  subjects,  which  have  been 

fully  treated  in  the  elementary  work,  is  greatly 

preferable  to  the  method  of  attempting  to  teach 

Additional    cvcry  thing  anew  :  for  there  must  necessarily  be 

stated,      much  that  is  common  ;  and  that  which  teaches 

no  new  principle,  or  indi  -ates  no  new  method  of 

application,  should  be  precisely  the  same  in  the 

higher  work  as  in  that  which  precedes  it. 

§  240.  To  vary  the  examples,  in  form,  without 

changing  in  the  least  the  principles   on  which 

A  contrary    they  are  worked,  and  to  arrange  a  thousand  such 

melnod  leads         ,,         .  ■,  ,  r         i  i         i 

to  confusion:  collcctions  uudcr  the  same  set  or  rules  and  sub- 
ject to  the  same  laws  of  solution,  may  give  a 
little  more  mechanical  facility  in  the  use  of 
figures,  but  will  add  nothing  to  the  stores  of 
arithmetical  knowledge.  Besides,  it  dehides  the 
learner  with  the  hope  of  advancement,  and  when 


CHAP.   II.]  ARITHMETIC CONCLUSION.  217 

he  reaches  the  end  of  his  higher  arithmetic,  he    it  misleads 

r-      1  1   •  ^^^  pupil: 

finds,  to  his  amazement,  that  he  has  been  con- 
ducted by  the  same  guides  over  the  same  ground 
through    a    winding    and    devious    way,    made      itcom- 

T         r  ■        1  1  -r        1  Plicntesthe 

Strange   by  lantastic  drapery:   whereas,  it  what      subject. 
was  new  had   been  classed  by  itself,  and  what 
was  known  clothed  in  its  familiar  dress,  the  sub- 
ject would  have  been  presented  in  an  entirely 
different  and  brighter  light. 


CONCLUDING     REMARKS. 

We  have  thus  completed  a  full  analysis  of  the    conciusioa 
language  of  figures,  and  of  the  construction  of 
numl)ers. 

We   have   traced  from  the  unit  one,   all   the      What 
numbers  or  arithmetic,  whether  integer  or  irac-       done, 
tional,  whether  simple  or  denominate.    We  have 
developed  the  laws  by  which  they  are  derived      Laws, 
from    this    common    source,   and    perceived    the 
connections  of  each  class  with  all  the  others. 

We  have  examined  that  concise  and  beautiful     Analysis 
language,  by  means  of  which  numbers  are  made      guage. 
available    in    rendering    the    results    of    science 
practically  useful ;  and  we  have  also  considered     Methods 

IT  1       1  r  1  •  1  ■  1   ■  °^  teaching 

the  best  methods  ot  teaching  this  great  subject    indicated. 

—the  foundation  of  all  mathematical  science —     import- 
ance of  the 
and  the  first  among  the  useful  arts.  subject. 


CHAP.  III.]  GEOMETRY.  319 


CHAPTER    III. 

OEOMETRY    DEFINED — THINGS     CF    WHICH    IT    TREATS — COMPARISON    AND    PROF- 

ERTIES     OF     FIGURES DEMONSTRATION  —  PROPORTION  —  SUGGESTIONS    FOB 

TEACHING. 

GEOMETRY. 

§  241.  Geometry   treats  of  space,   and  com-    Geometry. 
pares  portions  of  space  with  each  other,  for  the 
purpose  of  pointing  out  their  properties  and  mu- 
tual relations.     The  science  consists  in   the  de-   its  science, 
veiopment  of  all  the  laws  relating  to  space,  and 
is  made  up  of  the  processes  and  rules,  by  means 
of  which  portions  of  space  can  be  best  compared 
with  each  other.     The  truths  of  Geometry  are  a    ^^  ''"""^s- 
series  of  dependent  propositions,  and  may  be  di-     ^^  ^^^*^^ 
vided  into  three  classes  : 

1st.  Truths  implied  in  the  definitions,  viz.  that    1st.  Those 

.  .  impliod  in 

thmgs  do  exist,  or  may  exist,  corresponding  to    thedefini- 
the  words  defined.     For  example  :  when  we  say, 
"  A  quadrilateral  is  a  rectilinear  figure  having  four 
sides,"  we  imply  the  existence  of  such  a  figure. 

2d.   Self-evident,  or  intuitive  truths,  embodied  ^d.  Axioms, 
in  the  axioms  ;  and, 

3d.  Truths  inferred  from  the  definitions  and  3d.  Demon- 


220  MATHEMATICAL     SCIENCE.  [^BOOK  II, 

Btrative      axioms,  callcd  Demonstrative  Truths.     We  say 
that  a  truth  or  proposition  is  proved  or  demon- 

When  de- 
monstrated,  strated,   when,   by   a    course   of   reasoning,   it  is 

shown  to  be  included  under  some  other  truth  or 
proposition,  previously  known,  and  from  which 
is  said  io  follow ;  hence, 
Demonstra-  A  DEMONSTRATION  is  a  serics  of  logical  argu- 
ments, brought  to  a  conclusion,  in  which  the 
major  premises  are  definitions,  axioms,  or  prop- 
ositions already  established. 

Subjects  of       §  242.     Before  we  can  understand  the  proofs 
or  demonstrations  of  Geometry,  we  must  under- 
stand what    that   is   to  which  demonstration  is 
applicable :    hence,   the    first  thing  necessary  is 
to  form  a  clear  conception  of  space,  the  subject 
of  all  geometrical  reasoning.* 
Names  of        The  ucxt  stcp  is  to  givc  names  to  particular 
forms,      forms  or  limited  portions  of  space,  and  to  define 
these  names  accurately.    The  definitions  of  these 
names  are  the  definitions  of  Geometry,  and  the 
portions    of   space    corresponding    to    them    are 
Figures.      Called  Figurcs,  or  Geometrical  Magnitudes  ;    of 
Four  kinds,  wliicli  there  are  four  general  classes: 
First.  1st.  Lines; 

Second.  2d.  SurfacBS ; 

Third.  3(_].  Voluiiies ; 

Fourth.         4th.  Angles. 

*  Sections  80  to  84. 


CHAP.   III.]  GEOMETRY.  221 


§  243.  Lines  embrace  only  one  dimension  of      Jmibs. 
space,  viz.  length,  without  breadth  or  thickness. 
The  extremities,  or  limits  of  a  line,  are   called 
points. 

There  are  two  general  classes  of  lines — straight  Two  classes: 

..  ,  11-  A  -ii-  •  Straight  and 

hnes  and  curved  Imes.     A  straight  hne   is  one     curved. 
which  lies  in  the  same  direction  between    any 
two  of  its  points  ;  and  a  curved  line  is  one  which 
constantly  changes  its  direction  at  every  point. 
There  is  but  one  kind  of  straight  line,  and  that  is  one  kind  of 

/•  11         I  •       1    1  1  1    f>    •   •  T-i  1        straight  line 

lully  characterized  by  the  denmtion.  Jbrom  the 
definition  we  may  infer  the  following  axiom :  "  A 
straight  line  is  the  shortest  distance  between  two 
points."  There  are  many  kinds  of  curves,  of  many  or 
which  the  circumference  of  the  circle  is  the  sim- 
plest and  the  most  easily  described. 

§  244.    Surfaces  embrace  two  dimensions  of    Surfaces.- 
space,  viz.  length  and  breadth,  but  not  thickness. 
Surfaces,  like  lines,   are  also    divided  into    two    pianeand 
general  classes,  viz.  plane  surfaces  and  curved 
surfaces. 

A  plane  surface  is  that  with  which  a  straight     a  plane 
line,    any  how  placed,    and    having    two   points     ^'^'*'*' 
common  with  the  surface,  will  coincide  through- 
out  its  entire  extent.     Such   a   surface  is   per- 
fectly even,  and  is  commonly  designated  by  the     Perfectly 

even. 

term  "  A  plane."     A  large  class  of  the  figures 


222  MATHEMATICAL     SCIENCE.  [bOOK   II. 

Plane  Fig-  of  Gcometiy  are  but  portions  of  a  plane,  and  all 
such  are  called  plane  figures. 

§  245.  A  portion  of  a  plane,  bounded  by  three 
A  triangle,    straight  lincs,  is  called  a  triangle,  and  this  is  the 

the  most  sim- 
ple figure,    simplest  of  the  plane  figures.     There  are  several 

kind.5    of    triangles,   differing   from   each    other, 

however,   only  in    the    relative    values    of  their 

sides  and  angles.     For  example :  when  the  sides 

are  all  equal  to  each  other,  the  triangle  is  called 

Kinds  of  tri-  equilateral ;  when  two  of  the  sides  are  equal,  it 

"    ■      is  called  isosceles ;  and  scalene,  when  the  three 

sides  are  all  unequal.     If  one  of  the  angles  is  a 

right  angle,  the  triangle  is  called  a  right-angled 

triangle. 

§  246.  The  next  simplest  class  of  plane  figures 
comprises  all  those  which  are  bounded  by  four 

ftuadriiater-  Straight  lines,  and  are  called  quadrilaterals. 
There  are  several  varieties  of  this  class  : 

1st  species.  Ist.  The  mere  quadrilateral,  which  has  no 
special  mark,  called  a  trapezium. 

2d  species.  2d.  The  trapezoid,  which  has  two  sides  par- 
allel and  two  not  parallel ; 

3d  species.  3d.  The  parallelogram,  which  has  its  opposite 
sides  parallel  and  its  angles  oblique ; 

<th  species.  4th.  The  rectangle,  which  has  all  its  angles 
right  angles  and  its  opposite  sides  parallel ;  and, 


CHAP.   III. J  GiJOMETRY.  23.; 


5th.  The  square,  which  has  its  four  sides  equal   siu  species. 
to  each  other,  each  to  each,  and  its  four  angles 
right  angles. 

§  247.  Plane  figures,  bounded  by  straight  lines,  other  Poi>- 
having  a  number  of  sides  greater  than  four,  take 
names  corresponding  to  the  number  of  sides,  viz. 
Pentagons,  Hexagons,  Heptagons,  &c. 

§  248.    A    portion   of  a  plane   bounded  by  a     circles: 
curved  line,  all  the  points  of  which  are  equally 
distant  from   a  certain  point  within  called   the 
centre,  is  called  a  circle,  and  the  bounding  line 
is  called   the   circumference.     This   is   the  only  the  circum 
curve  usually  treated  of  in  Elementary  Geometry. 

§  249.    A  curved  surface,  like  a  plane,  em-  curved  su^ 

fices: 

braces  the  two  dimensions  of  length  and  breadth. 

It  is  not  even,  like  the  plane,  throughout  its  whole 

extent,  and   therefore  a  straight  line  may  have  their  proper 

two  points  in  common,  and  yet  not  coincide  with 

it.     The  surface  of  the  cone,  of  the  sphere,  and 

cylinder,  are  the  curved   surfaces  treated  of  in 

Elementary  Geometry. 

§  250.  A  volume  is  a  limited  portion  of  space, 
combining-  the  three  dimensions  of  length,  breadth, 
and  thickness.     Volumes  are  divided  into  three      Three 

cla(>«eB. 

classes: 


^24  MATHEMATICAL     SCIENCE,  [bOOKIT. 


1st  class.         1st.  Those  bounded  by  planes  ; 
2d  class.         2d.  Those  bounded  by  plane  and  curved  sur- 
faces ;  and, 
3d  class.  3d.  Those  bounded  only  by  curved  surfaces. 

What  figures      The    first    class    embraces    the   pyramid    and 

fill]  in  each  .  .   ,        ,      .  ,  .      . 

class.  pnsm  With  their  several  varieties  ;  the  second 
class  embraces  the  cylinder  and  cone ;  and  the 
third  class  the  sphere,  together  with  others  not 
generally  treated  of  in  Elementary  Geometry. 

Magnitudes  §  251.  We  have  now  named  all  the  geomet- 
rical magnitudes  treated  of  in  elementary  Ge- 

What  they  oiiietry.  They  are  merely  limited  portions  of 
space,  and  do  not,  necessarily,  involve  the  idea 

A  sphere,  of  matter.  A  sphere,  for  example,  fulfils  all  the 
conditions  imposed  by  its  definuions,  without  any 
reference  to  what  may  be  within  the  space  en- 
Need  not  be  closed  by  its  surface.     That  space  may  be  oc- 

malerial. 

cupied  by  lead,  iron,  or  air,  or  may  be  a  vacuum, 
without  at  all  changing  the  nature  of  the  sphere, 
as  a  geometrical  magnitude. 

It  should  be  observed   that   the  boundary  or 

Uoiindaries  limit  of  a  geometrical  magnitude,  is  another  geo- 
metrical magnitude,  having  one  dimension  less. 
For  example:  the  boundary  or  limit  of  a  volume, 

Examples.  Avhicli  luis  three  dimensions,  is  always  a  surface 
"which  has  but  two;  the  limits  or  boundaries  of 


CHAP,   III.]  GEOMETRY.  225 

all  surfaces  are  lines,  straight  or  curved  ;  and  the 
extremities  or  limits  of  lines  are  points. 


§  252.    We  have  now  named  and  shown  the     subjects 

named. 

nature   of  the  things   which  are  the  subjects  of 


Science  of 
Geometry. 


Elementary  Geometry.  The  science  of  Ge- 
ometry is  a  collection  of  those  connected  pro- 
cesses by  which  we  determine  the  measures, 
properties,  and  relations  of  these  magnitudes. 

COMPARISON  OF  FIGURES  AVITH  UNITS  OF  MEASURE, 

§  253.  We  have  seen  that  the  term  measure     Measure, 
implies  a  comparison  of  the  thing  measured  with 
some  known  thing  of  the  same  kind,  regarded 
as  a  standard ;  and  that  such  standard  is  called 
the  unit  of  measure.*     The  unit  of  measure  for  unitofmea* 

ure 

lines  must,  therefore,  be  a  line  of  a  known  length  :    For  Linea, 

a  foot,  a  yard,  a  rod,  a  mile,  or  any  other  known 

unit.     For  surfaces,   it  is  a  square   constructed  ForSurfacea, 

on   the  linear  unit  as  a  side:  that   is,  a  square    a  Square;. 

foot,  a  square  yard,  a  square  rod,  a  square  mile ; 

that  is,  a   square  described  on   any  known  unit 

of  length. 

The  unit  of  measure,  for  volumes,  is  a  volume,  ForSoMs. 
and  therefore  has  three  dimensions.     It  is  a  cube     a  Cube, 

*  Section  94. 
15 


226  MATHEMATICAL     SCIENCE,  [boOK  II. 


constructed  on  a  linear  unit  as  an  edge,  or  on 

the  superficial  unit  as  a  base.     It  is,  therefore, 

a  cubic    foot,   a  cubic   yard,  a  cubic    rod,   &c. 

Three  units  Hcnce,   there   are   three  units  of  measure,  each 

0  measur  .  ^j^g^^-j^^g  jj^  j^j^^^  from  the  Other  two,  viz.  a  known 

A  Line,      length  foT  the  measurement  of  Hues ;  a  known 
A  Square,     squarc  for  the  measurement  of  surfaces ;  and  a 
A  Cube,     known   cube   for   the   measurement  of  volumes. 
Contents:   The  mcasurc  or  contents  of  any  magnitude,  be- 
how  ascer-  longing  to  either  class,  is  ascertained  by  finding 
how   many   times   that    magnitude   contains   its 
unit  of  measure. 

§  254.  ■  In  the  fourth  class  of  the  Geometrical 
magnitudes,  there  are  several  varieties.  First, 
the  inclination  of  lines  to  each  other;  2d,  of 
planes;  and  3d,  the  space  included  by  three  or 
more  planes  meeting  at  a  point.  In  Geometry, 
Angles:     the  right  angle  is  the  simplest  unit, — in   Trig- 

theii-  unit. 

onometry,  the  degree,  with   its   subdivisions. 

§  255.  We  have  dwelt  with   much  detail  on 

the  unit  of  measure,   because    it    furnishes    the 

Importance  only  basis  of   estimating    quantity.       The   Con- 
or the  unit  of 

measure:  ccptiou  of  iiuniber  and  space  merely  opens  to 
the  intellectual  vision  an  unmeasured  field  of 
investigation  and  thought,  as  the  ascent  to  the 
summit  of  a  mountain  presents  to    the   eye  a 


CHAP.    III.]  GEOMETRY.  227 


wide  and  unsurveyed  horizon.     To  ascertain  the  Ppace  indefl- 

t      ■    ,  r     1  •  r      •  IT  ri         nite  without 

height  01  the  point  ot  view,  the  diameter  oi  the        j,.    • 

surrounding  circular  area    and   the   distance   to 

any  point  which  may  be  seen,  some  standard  or 

unit    must    be    known,  and   its  value  distinctly 

apprehended.     So,  also,  number  and  space,  which 

at  fii-st  fill  the  mind  with  vague  and  indefinite   "»d  always 

measured 

conceptions,  are  to  be  finally  measured*  by  units       by  \u 
of  ascertained  value. 


§  256.    It   is  found,   on   careful  analysis,   that   Every  mim- 
every  number  may  be  referred  to  the  unit  one,    JfeiTeaio 
as  a  standard,  and  when  the  signification  of  the  '"^'^ "'"'  ^'^ 
term  one  is  clearly  apprehended,  that  any  num- 
ber, whether  lars;e  or  small,  whether  integral  or 
fractional,  may  be  deduced  from  the  standaixi  by  ^ 

an  easy  and  known  process. 

In  space,   also,  which  is   indefinite   in  extent,      space? 
and  exactly  similar  in  all  its  parts,  the  faculties 
of  the  mind  have   established   ideal  boundaries,     its  ideal 
These  boundaries  give  rise  to  the  geometrical    """  '^"^ 
magnitudes,  each  of  which  has  its  own  unit  of 
measure ;  and  by  these  simple  contrivances,  we 
measure  space,  even  to  the  stars,  as  with  a  yard- 
stick. 


§  257.  We  have,  thus  far,  not  alluded  to  the 
difficulty  of  determining  the  exact  length  of  that 


228  MATHEMATICAL     SCIENCE.  [boOK   II 

Conception    which  wc  regard  as  a  standard.     We  are  pre- 

of  the  unit  of  i         •   i  •  i  i  i         i  i       i  -       • 

measui-e:  scnted  With  a  given  length,  and  told  that  it  is 
called  a  foot  or  a  yard,  and  this  being  usually 
done  at  a  period  of  life  when  the  mind  is  satis- 
fied with  mere  facts,  we  adopt  the  conception 
At  first,  a     of  a  distance  corresponding  to  a  name,  and  then 

■  sion.^"^  ^y  multiplying  and  dividing  that  distance  we 
are  enabled  to  apprehend  other  distances.  But 
this  by  no  means  answers  the  inquiry,  What  is 
the  standard  for  measurement  ? 

Standard  of      The  cominoii  standards  of  measurement,  1  yard. 

measure- 
ment.      1  foot,  Avitli  their  multiples  and  subdivisions,  are 

derived  from  the  English  Exchequer  and  the  laws 

of  Great   Britain.      The   one   common    standard 

from  "vvhich  they  are  all  deduced  is  the  English 

A  brass  rod.  yard.  The  positive  standard  yard,  is  a  brass  rod 
of  the  year  1601,  deposited  in   the  British  Ex- 

A)i  weights  cliequep.     All  the  weiglits  and  measures  in  the 

and  meas-     t       •       i     o  t      •       t     p 

iirescome   United  States,  in  general  use,  are  derived  from 

this  standard.     Besides  this  standard,  there  is  yet 

Metre  nii:o  a  another,  in  very   general   use,  and   consequently 

standard. 

another  system  of  Weights  and  Measures,  known 

as  the  Metric  system  of  France. 

Primary         The  primary  base  of  this   system,  for  all  de- 
base. 

nominations   of   Weights   and   Measures,    is   the 

one-ten-millionth  part  of  the  distance  from  the 
equator  to  the  pole,  measured  on  the  meri- 
dian. 


CHAP.    III.]  GEOMETRY.  239 

It  is  culled  a  Metre,  and  is  equal  to  39.37    a  metre, 
inches,  very  nearly. 

The  imperial  yard  has  also  been  referred  to 
an  invariable  standard,  viz.  the  distance  between  invariable 
the  axis  of  suspension  and  the  centre  of  oscilla-  ^'^"  ^'  ' 
tion  of  a  pendulum  which  shall  vibrate  seconds 
in  vacuo,  in  London,  at  the  level  of  the  sea.  This 
distance  is  found,  and  declared  to  be,  39.1393 
imperial  inches  ;  that  is,  3  imperial  feet  and  3.139 
inches. 

§258.   The  standard  unit  of  length  is  not  only     Unit  of 

lcn<;th 

important,  as  aflFordmg  a  basis  for  all  measure-  important, 
ments  of  surface   and  capacity,  but  is  also   the     „  > 
element  from  which  we  deduce  the  unit  of  weight,     weight 
The  weight  of  a  cubic  foot  of  pure  rain-water,  is  conies  from 
divided  into  one  thousand  equal  parts,  and  each        ^^' 
part  is  called  one  ounce.     Sixteen  of  these  ounces 
make  the  pound  avoirdupois,  which  is  our  com- 
mon unit  of  Aveight.     Hence,  the  existing  weights  Weights  and 
and  measures  of  the  United  States,  are  derived     EugUsh. 
from   the   English   Excliequer   and  the  laws  of 
Great  Britain. 


Equal 


§  259.   Two  geometrical  figures  are  said  to  be 
equal,  when  they  contain  the  same  unit  of  meas-     figures; 
ure  an  equal  number  of  times.     Two  figures  are 
said  to  be  equal  in  all  their  parts,  each  to  each, 


and 


230  MATHEMATICAL    SCIENCE.  [iJOOK   II. 

equal  in  all  when  they  can  be  so  applied  to  each  other  as  to 

eirpars.  ^^jj^^^jg   throiighont   their  whole   extent.     The 

term  equal,  is  thus  used  in  Geometry  in  the  same 

sense  in  which  it  is  used  in  Arithmetic  and  in 

DiflFerence  Analysis:  viz.  to  denote  the  relation  between  two 
equal  quantities  each  of  which  contains  the  same  unit 
an  equal  number  of  times.  If  two  geometrical 
magnitudes  can  be  applied,  the  one  to  the 
other,  so  as  to  coincide,  they  are  not  only  equal 
in  measure,  but  each  part  of  the  one  is  equal  to 

equal  in  all  ^  corresponding  part  of  the  other:  hence,  they 
are  said  to  be  equal  in  all  their  parts. 

PROPERTIES     OF     FIGURES. 

Property  of        §  2G0.  A  property  of  a  figure  is  a  mark  com- 
mon to  all  figures  of  the  same  class.     For  exam- 
^uadriiater-  pie  :  if  the  class  be  "  Quadrilateral,"  there  are  two 

als. 

very  obvious  properties,  common  to  all  quadri- 
laterals, besides  the  one  which  characterizes 
the  figure,  and  by  which  its  name  is  defined, 
viz.  that  it  has  four  angles,  and  that  it  may 
be   divided   into  two   triangles.     If  the  class  be 

Paraiieio-    "  Parallelogram,"    there    are    several    properties 

^'^"™*       common  to   all   parallelograms,    and  which    are 

subjects  of   proof ;    such    as,   that  the  opposite 

sides  and  angles  are  equal ;  the  diagonals  divide 

each  other  into  equal  parts,  &c.     If  the  class  be 

triangle:  "Triangle,"  there  are  many  properties  common 
to  all  triangles,  besides   the  characteristic    that 


CHAP,   irr.]  GEOMETRY.  231 


they  have  three  sides.     If  the  class  be  a  par-    Equuatemi, 
ticular  kind  of  triangle,  such  as  the  equilateral,    isosceles, 
isosceles,  or  right-angled  triangle,  then  each  class  Right-angieJ. 
has  particular  properties,  common  to  every  indi- 
vidual of  the  class,  but  not  common  to  the  other 
classes.      It  is  important,  however,   to   remark,  Every  prop 

erty  which 

that  every  property  which  belongs  to  "  triangle,"  belongs  to  a 

,     ,  .,,  .  genus  will  be 

regarded   as    a  genus,   will    appertain    to    every   common  to 
species    or   class   of  triangle ;    and   universally,    ^^'^■gg?'^ 
every  property  which  belongs   to  a  genus  will 
belong   to    every   species  under  it ;    and    every 
property  which    belongs   to   a   species  will    be- 
long to  every  class  or  subspecies  under  it;  and  also  to  every 
every  property  which  belongs  to  one  of  a  sub-    ^"^^p'^<='^^. 

•I     i^        f^        -I  o  and  to  every 

species  or  class  will  be   common  to  every  indi-    '"dividual, 
vidual  of  the  class.     For  example  :  "  the  square    Examples, 
on  the  hypothenuse  of  a  right-angled  triangle  is 
equal   to  the   sum  of  the  squares  described  on 
the   other   two    sides,"   is   a   proposition   equally 
true  of  every  right-angled  triangle:  and  "every 
straight  line    perpendicular  to    a   chord,   at   the      circle, 
middle  point,  will  pass  through   the  centre,"   is 
equally  true  of  all  circles. 


MARKS  OF  WHAT  MAY  BE  PROVED. 

§  201.  The  characteristic  properties  of  every  characteri.* 
geometrical  figure  (that  is,  those  properties  with-   ''"t^^^"' 


232  MATHEMATICAL     SCIENCE.  [bOOK  11. 

out  which  the  figures  could  not  exist),  are  given 

■  ■  in  the  definitions.     How  are  we  to  arrive  at  all 

the    other   properties    of    these    figures  ?      The 

propositions  implied  in  the  definitions,  viz.  that 

Marks:      things  Corresponding  to  the  words  defined  do  or 

may  exist  with  the  properties  named  ;  and   the 

Of  what  may  self-cvident  propositions  or  axioms,  contain  the 

be  proved.  /-        i  i  i     i 

only  marks  of  what  can  be  proved  ;  and   by   a 
How  ex-     gi^iifui  combination  of  these  marks  we  are  able 

tended. 

to  discover  and  prove  all  that  is  discovered  and 
proved  in  Geometry. 

Definitions   and   axioms,  and   propositions  de- 
'^^^°^      duced  from    them,   are   maior  premises   in  each 

Premiss.  •>         ^ 

The  science:  new  demonstration  ;  and  the  science  is  made  up 

in  what  it         ^      .  i  i     r        i^    •        •  c 

consists.  01  the  processes  employed  tor  bringing  uniore- 
seen  cases  under  these  known  truths ;  or,  in  syl- 
logistic language,  for  proving  the  minors  neces- 
sary to  complete  the  syllogisms.  The  marks 
being  so  few,  and  the  inductions  which  furnish 
them  so  obvious  and  familiar,  there  would  seemi 
to  be  very  little  difficulty  in  the  deductive  pro- 
cesses which  follow.  The  connecting  together 
of  several  of  these  marks  constitutes  Deductions, 
Geometry,    or  Trains  of  Reasoning ;  and  hence,  Geometry 

a  Deductive    .  i-v     i       ,  •         ci    • 

Science.     ^^  ^  Dcductivc  ocience. 


CUAP.  in.] 


GEOMETRY. 


233 


Enunciatiuu. 


DEMONSTRATION. 

§  2G2.  As  a  first  example,  let  us  take  the  first 
proposition  in  Legendre's  Geometry  : 

"If  a  straight  line  meet  another  straight  line,   Pruposition 
the  sum  of  the  two  adjacent  angles  will  be  equal 
to  two  right  angles." 

Let  the  straight  line  DC 
meet  the  straight  line  AB 
at  Ihe  point  C,  then  will  the 
angle  ACD  plus  the  angle 
DCB  be  equal  to  two  right     ^  ^  ■' 

angles. 

To  prove  this  proposition,  we  need  the  defini 
tion  of  a  right  angle,  viz. : 

When  a  straight  line  AB  B 

meets  another  straight  line 
CD,  so  as  to  make  the  ad- 
jacent angles  BAG  and 
BAD   equal  to   each  other, 

each  of  those  angles  is  called  a  right  angle,  and 
the  line  AB  is  said  to  he  perpendicular  to  CD. 

We  shall  also  need  the  2d,  3d,  and  4th  axioms, 
I'or  inferring  equality,*  viz.  : 

2.   Things  which  are  equal  to  the  same  thing 
are  equal  to  each  other. 


Things 

necessary  to 

prove  it. 


D     Deflnitions. 


Axioms. 


Second. 


*  Section  109. 


234 


MATHEMATICAL     SCIENCE. 


[book  11 


3.  A   whole  is    equal    to  the   sum   of  all  its 
parts. 

4.  If  equals    be   added    to    equals,   the  sums 
will  be  equal. 

Now  before  these  formulas  or  tests  can  be  ap- 

E        D 


unetobe    plied,  it  is  nccessaiy  to  sup- 
drawn. 

pose  a  straight  line  CE  to  be 
drawn  perpendicular  to  AB 
at  the  point  C  :  then  by  the 
definition   of  a   right    angle, 


Proof: 


Conclusion. 


Ita  bases. 


First. 


A  C  B 

the  angle  ACE  will  be  equal  to  the  angle  ECB. 

By  axiom  3rd,  we  have, 

ACD  equal  to  ACE  plus  ECD:  to  each  of 
these  equals  add  DCB ;  and  by  the  4th  axiom 
we  shall  have, 

ACD  plus  DCB  equal  to  ACE  plus  ECD  plus 
DCB  ;  but  by  axiom  3rd, 

ECD  plus  DCB  equals  ECB:  therefore  by 
axiom  2d, 

ACD  plus  DCB  equals  ACE  plus  ECB. 

But  by  the  definition  of  a  right  angle, 

ACE  plus  ECB  equals  two  right  angles  :  there- 
fore, by  the  2d  axiom, 

ACD  plus  DCB  equals  two  right  angles. 

It  will  be  seen  that  the  conclusiveness  of  tiie 
proof  results, 

1st.  From  the  definition,  that  ACE  and  ECB 
are  equal  to  each  other,   and  each  is  called  a 


CHAP.  III.]  GEOMETRY.  235 

right-angle  :  consequently,  their  sum  is  equal  to 
two  right  angles  ;  and, 

2dly.  In  showing,  by  means  of  the  axioms,  that     Secca-i. 
ACD  plus  DCB  equals   ACE  plus  ECB;    and 
then  inferring  from  axiom  2d,  that,  ACD  plus 
DCB  equals  two  right  angles. 

§203.  The  difficulty  in  the  geometrical  rea-  Difficulties  in 

the  demon- 

soning  consists  mainly  in  showing  that  the  prop-     strations. 
osition  to  be  proved  contains  the   marks  which 
prove   it.     To    accomplish  this,   it  is  frequently 
necessary  to  draw  many  auxiliary  lines,  forming    Auxiliaries 

n  11  I  •    1  11  necessary. 

new  figures  and  angles,  which  can  be  shown  to 
possess  marks  of  these  marks,  and  which  thus 
become  connecting  links  between  the  known  connecting 
and  the  unknown  truths.  Indeed,  most  of  the 
skill  and  ingenuity  exhibited  in  the  geometrical 
processes  are  employed  in  the  use  of  these  auxil- 
iary means.  The  example  above  affords  an  illus- 
tration. We  were  unable  to  show  that  the  sum  How  used. 
of  the  two  angles  possessed  the  mark  of  being 
equal  to  two  right  angles,  until  we  had  drawn  a 
perpendicular,  or  supposed  one  drawn,  at  the 
point  where  the  given  lines  intersect.  That  be- 
ing done,  the  two  right  angles  ACE  and  ECB  collusion. 
were  formed,  which  enabled  us  to  compare  the 
sum  of  the  angle  ACD  and  DCB  with  two  right 
angles,  and  thus  we  proved  the  proposition. 


236 


MATHEMATICAL     SCIENCE.  [BOOK    II. 


niagrara. 


Principles 
necessary. 


Proposition.      §  2G4.   As  a  secoiid  example,  let  us  take  tlie 

following  proposition : 
Eiumciation.      if  i'i^o  stvaiglit  Uiies  intersect  each  otlier,  the 
opjjosite  or  vertical  angles  will  he  equal. 

Let  the  straight  line  AB    ^^  ^  j) 

intersect  the  straight  line 
ED  at  the  point  C  :  then 
will  the  angle  ACD  be 
equal  to  the  opposite  an- 
gle ECB ;  and  the  angle  ACE  equal  to  the  an- 
gle DCB. 

To  prove  this  proposition,  we  need  the  last 
proposition,  and  also  the  2d  and  5th  axioms,  viz. : 

"  If  a  straight  line  meet  another  straight  line, 
the  sum  of  the  two  adjacent  angles  will  be  equal 
to  two  rio-ht  angles." 

"  Things  which  are  equal  to  the  same  thing 
are  equal  to  each  other." 

"  If  equals  be  taken  from  equals,  the  remain- 
ders will  be  equal." 

Now,  since  the  straight  line  AC  meets  the 
straight  line  ED  at  the  point  C,  we  have, 

ACD  plus  ACE  equal  to  two  right  angles. 

And  since  the  straight  line  DC  meets  the 
straight  line  AB,  we  have, 

ACD  plus  DCB  equal  to  two  right  angles : 
hence,  by  the  second  axiom, 

ACD  plus  ACE  equals  ACD  plus  DCB:  ta- 


Proof. 


CHAP.   III.]  GEOMETRY.  23' 


kino;  from    each   the   common    angle    ACD,   we  conclusion, 
know    from    the    fifth    axiom    that   the   remain- 
ders   will    be    equal ;    that    is,    the    angle    ACE 
equal  to  the  opposite  or  vertical  angle  DCB. 

§  265.    The  two  demonstrations  given  above 
combine  all  the  processes  of  proof  employed  in   Demonstra- 
every  demonstration  of  the  same  class.     When  "o°»  =«"«"■"'• 
any  new  truth  is  to  be  proved,  the  known  tests 
of    truth    are    gradually    extended    to    auxiliary  useof  auxii- 
quantities    havino-  a    more  intimate    connection    »'^>'i"'^"* 

^  °  lies. 

with  such  new  truth  than  existed  between  it  and 
the  known  tests,  until  finally,  the  known  tests, 
through  a  series  of  links,  become  applicable  to 
the  final  truth  to  be  established  :  the  interme- 
diate processes,  as  it  were,  bridging  over  the 
space  between  the  known  tests  and  the  final 
truth  to  be  proved. 

§  26G.  There  are  two  classes  of  demonstra-  Direct  dem 
lions,  quite  different  from  each  other,  in  some 
respects,  although  the  same  processes  of  argu- 
mentation are  employed  in  both,  and  although 
the  conclusions  in  both  are  subjected  to  the 
same  logical  tests.     They  are  called  Direct,  or    ^ 

^  •'  Negative, 

Positive   Demonstration,   and  Negative  Demon-        o«" 

Reductio  ad 

stration,  or  the  Reductio  ad  Absurdum.  Absuidum. 


238  MATHEMATICAL     SCIENCE.  [bOOK  IL 


Difference.         §  267.      The    maiii    differences   in   the    two 
methods  are  these  :  The  method  of  direct  demon- 
Direct  Dom-  stration  rests  its  arguments  on  known  and  ad- 

onslration. 

mitted    truths,  and   shows    by   logical    processes 

that  the  proposition  can  be  brought  under  some 

previous  definition,  axiom,  or  proposition  :  while 

Negative     the  negative  demonstration  rests  its  arguments 

Demonstra- 
tion,      on  an  hypothesis,  combines  this  with  known  pro- 
positions, and  deduces  a  conclusion  by  processes 

Conclusion:  strictly  logical.  Now  if  the  conclusion  so  de- 
duced agrees  with   any  known  truth,   we    infer 

With  what   ^^^^  ^j^g  hypothesis,  (which  was  the  only  link  in 

compared.  ^  i  \ 

the  chain  not  previously  known),,  was  true ;  but 
if  the  conclusion  be  excluded  fro.Ti  the  truths 
previously  established ;  that  is,  if  it  he  opposed 
to  any  one  of  them,  then  it  follows  thai  the  hy- 
pothesis, being  contradictory  to  such  truth,  must 
Determines   ^^  false.     In  the  negative  demonstration,  there- 

whether  the 

hypothesis  is  fore,  the  conclusion  is  compared  with  the  truths 

true  or  false.  ... 

known  antecedently  to  the  proposition  m  ques- 
tion :  if  it  agrees  with  any  one  of  them,  the  hy- 
pothesis is  correct ;  if  it  disagrees  with  any  one 
of  them,  the  hypothesis  is  false. 

Proof  by        g  2G8.    We  will  give  for  an  illustration  of  this 

Negative 

uemonstra-  mcthod,  Propositiou  XVII.  of  the  First  Book  of 
Legendre :  "  When  two  right-angled  triangles 
have  the  hypothenuse  and  a  side  of  the  one  equal 


CHAP.   III.]  GEOMETRY.  239 


to  the  hypothenuse  and  a  side  of  the  other,  each  Enunciatiou 
to  each,  the  remaining  parts  will  be  equal,  each  to 
each,  and  the  triangles  themselves  will  be  equal." 

In  the  two  right-angled  triangles  BAG  and 
EDF  (see  next  figure),  let  the  hypothenuse  AC  Enunciation 
be  equal  to  DF,  the  side  BA  to  the  side  ED :  ^  ^  ''"'^' 
then  will  the  side  BC  be  equal  to  EF,  the  angle 
A  to  the  angle  D,  and  the  angle  C  to  the  ano-le  F. 
To  prove  this  proposition,  we  need  the  follow- 
ing, which  have  been  before  proved  ;  viz. : 

Prop.  X.  (of  Legendre).   "When  two  triangles     previous 
liave  the  three  sides  of  the  one  equal  to  the  three  "''"''^  °*^*** 

'  sary. 

sides  of  the  other,  each  to  each,  the  three  an- 
gles will  also  be  equal,  each  to  each,  and  the 
triangles  themselves  will  be  equal." 

Prop.    V.     "  When    two    triangles    have    two  Proposition 
sides  and   the  included  angle  of  the  one,  equal 
to  two  sides  and  the  included  angle  of  the  other, 
each  to  each,  the  two  triangles  will  be  equal." 

Axiom   I.    "  Things   which   are    equal   to    the     Axioms, 
same  thing,  are  equal  to  each  other." 

Axiom   X.    (of  Legendre).    "  All  right  angles 
are  equal  to  each  other." 

Prop.  XV.  "  If  from  a  point  without  a  straight  Proposition, 
line,  a  perpendicular  be  let  fall  on  the  line,  and 
oblique  lines  be  drawn  to  different  points, 

1st.  "The  perpendicular  will  le  shorter  than 
any  oblique  line ; 


340 


MATHEMATICAL     SCIENCE, 


[book  H. 


c  a 


Constriictiou 
of  the  figure. 


2d.  "Of  two  oblique  lines,  drawn  at  pleasure, 
that  which  is  farther  from  the  perpendicular  will 
be  the  longer." 
Now  the  two  sides  BC  and 
neginning  cf  EF  are  either  equal  or  un- 

the  demon- 
stration,     equal.       If    they   are  equal, 

then  by  Prop.  X.  the  remain- 
ing parts  of  the  two  trian- 
gles are  also  equal,  and  the  triangles  themselves 
are  equal.  If  the  two  sides  are  unequal,  one  of 
them  must  be  greater  than  the  other :  suppose 
BC  to  be  the  greater. 

On  the  greater  side  BC  take  a  part  BG,  equal 
to  EF,  and  draw  AG.  Then,  in  the  two  trian- 
gles BAG  and  DEF  the  angle  B  is  equal  to  the 
angle  E,  by  axiom  X  (Legendre),  both  being 
right  angles.  The  side  AB  is  equal  to  the  side 
DE,  and  by  hypothesis  the  side  BG  is  equal  to  the 
side  EF :  then  it  follows  from  Prop.  V.  that  the 
side  AG  is  equal  to  the  side  DF.  But  the  side 
DF  is  equal  to  the  side  AC  :  hence,  by  axiom  I, 
the  side  AG  is  equal  to  AC.  But  the  line  AG 
cannot  be  equal  to  the  line  AC,  having  been 
shown  to  be  less  than  it  by  Prop.  XV. :  hence, 
the  conclusion  contradicts  a  known  truth,  and  is 
thei'efore  false  ;  consequently,  the  supposition  (on 
which  the  conclusion  rests),  that  BC  and  EF  are 
unequal,  is  also  false  ;  therefore,  they  are  equal 


Uemonstra- 
tion. 


CHAP.    III.]  GEOMETRY.  241 


§  2G9.  It  is  often  supposed,  though  erroneous-     Negative 
ly,  that  the  Negative  Demonstration,  or  the  dem-       ^o^. 
onstration  involving  the  "  reductio  ad  absurdurn," 
is  less  conclusive  and  satisfactory  than  direct  or    conclusive, 
positive  demonstration.     This  impression  is  sim- 
ply the  result  of  a  want  of  proper  analysis.     For 
example  :  in  the  demonstration  just  given,  it  was     Reaaojis. 
proved  that  the  two  sides   BC  and  EF  cannot 
be  unequal;  for,  such  a  supposition,  in  a  logi- 
cal argumentation,  resulted  in  a  conclusion  di-    Conclusion 

,  ,  ,  ,  ,  ,.  corresponds 

rectly opposed  to  a  known  truth;  and  as  equality  to, oris op- 
and   inequality  are  the  only  general  conditions     posed  to 

'  •'  •^     ^  known  truth. 

of  relation  between  two  quantities,  it  follows 
that  if  they  do  not  fulfil  the  one,  they  must  the 
other.  In  both  kinds  of  demonstration,  the 
premises  and  conclusion  agree  ;  that  is,  they  are  Agreement 
both  true,  or  both  false ;  and  the  reasoning  or 
argument  in  both  is  supposed  to  be  strictly  logi- 
cal. 

In  the  direct  demonstration,  the  premises  are 
known,    being   antecedent    truths ;    and    hence, 
the  conclusion  is  true.     In  the  negative  demon-  Differencesin 
stration,  one  element   is    assumed,  and  the  con-       ^^^ 
elusion  is  then  compared  with  truths  previously 
established.     If  the  conclusion  is  found  to  agree 
with  any  one    of  these,  we  infer  that    the  hy-     when  the 
pothesis  or  assumed  element  is  true  ;  if  it  con-    ^^'"rur'^ 
tradicts  any  one  of  these  truths,  we  infer  that 

16 


243 


MATHEMATICAL     SCIENCE. 


[flOOK   II. 


When  false,  the  assumed  element  is  false,  and  hence  that  its 
opposite  is  true. 


Measured : 
tin  significa- 
tion. 


§  270.  Having  explained  the  meaning  of  the 
term  measured,  as  applied  to  a  geometrical  mag- 
nitude, viz.  that  it  implies  the  comparison  of  a 
magnitude  with  its  unit  of  measure ;  and  having 
also  explained  the  signification  of  the  word  Prop- 
Generai     erty,  and  the  processes  of  reasoning  by  which, 

Remarks. 

in  all  figures,  properties  not  before  noticed  are 
inferred  from  those  that  are  known  ;  we  shall 
now  add  a  few  remarks  on  the  relations  of  the 
geometrical  figures,  and  the  methods  of  compar- 
incT  them  with  each  other. 


PKOPORTION     OF     FIGURES. 

Proportion.  §  271.  Proportion  is  the  relation  which  one 
geometrical  magnitude  bears  to  another  of  the 
same  kind,  with  respect  to  its  being  greater  or 
less.    The  two  magnitudes  so  compared  are  called 

Its  measure,  terms,  and  the  measure  of  the  proportion  is  the 
quotient  which  arises  from  dividing  the  second 
term  by  the  first,  and  is  called  their  Ratio.     Onlv 

Ratio.  J  '  •/ 

uuautiiies  of  quantities   of  the   same   kind   can   be  compared 
thosamt     together,  and  it  follows  from  the  nature  of  the 

kind  com  ° 

pared.  relation  that  the  quotient  or  ratio  of  any  two 
terms  will  be  an  abstract  number,  whether  the 
terms  themselves  be  abstract  or  concrete 


CHAP.  III.]  GEOMETRY.  243 


§  273.   The  term  Proportion  is  defined  by  most  Proponion: 
autliors,   "An  equality  of   ratios    between  four 
numbers   or  quantities,  compared  together  two 
and  two."     A    proportion   certainly  arises   from 
such  a  comparison  :  thus,  if 

B     D      ^  .         , 

-.-  =  -77  ;    then,  Example, 

A      C 
A  :  B  :  :  C  :  D 

IS  a  proportion. 

But  if  we  have  two  quantities  A  and  B,  which   True  defioi- 

tioD. 

may  change  their  values,  and  are,  at  the  same 
time,  so  connected  together  that  one  of  them 
shall  increase  or  decrease  just  as  many  times  as 
the  other,  their  ratio  will  not  be  altered  by  such 
chanai;es ;  and  the  two  quantities  are  then  said  ^'^°i'^"P"y~ 
to  be  in  proportion,  or  proportional.  '''^ 

Thus,  if  A  be  increased  three  times  and  B 
three  times,  then, 

3B_A 
3A~B' 

that  is,  3  A  and  3  B  bear  to  each  other  the  same 
proportion  as  A  and  B.  Science  needed  a  gen-  Term  need- 
eral  term  to  express  this  relation  between  two 
quantities  which  change  their  values,  without 
altering  their  quotient,  and  the  term  "propor- 
tional," or  "in  proportion,"  is  employed  for  that  Howiiaod 
purpose. 


244  MATHEMATICAL     SCIENCE.  [bOCK    II. 


Eeasonsfor       As  somc  apologj  for  the  modification  of  the 

mc  I  caion.  jg^j^j^^j^  ^^  proportion,  which  has  been  so  long 
accepted,  it  may  be  proper  to  state  that  the  term 
has  been  used  by  the  best  authors  in  the  exact 

Use  of  the  sense  here  attributed  to  it.  In  the  definition  of 
the  second  law  of  motion,  we  have,  "  Motion, 
or  change  of  motion,  is  proportional  to  the  force 
impressed  ;"*  and  again,  "  The  inertia  of  a  body 
is  proportioned  to  its  weight."!  Similar  exam- 
ples may  be  multiplied  to  any  extent.     Indeed, 

Symbol  used  there  is  a  symbol    or  character  to  express    the 

(o  represent 

proportion,  relation  between  two  quantities,  when  they  un- 
dergo changes  of  value,  without  altering  their 
ratio.  That  character  is  oc,  and  is  read  "  pro- 
portional to."  Thus,  if  we  have  two  quantities 
denoted  by  A  and  B,  written. 

Example.  A  CX  JD, 

the  expression  is  read,  "  A  proportional  to  B." 
Another  kind       §  273.   There  is  yet  another  kind  of  relation 

<)f  proper-  ,  .    ,  •         1  •    •  A  J 

tion.  which  may  exist  between  two  quantities  A  and 
B,  which  it  is  very  important  to  consider  and 
understand.  Suppose  the  quantities  to  be  so 
connected  with  each  other,  that  when  the  first 
is  increased  according  to  any  law  of  change,  the 
second  shall  decrease  according  to  the  same  law  j 
and  the  reverse. 

»  Olmsted's  Mechanics,  p.  23.  f  Ibid.  p.  23. 


CHAP.  III. 


GEOMETRY, 


245 


First 
Example 


For  example  :  the  area  of  a  rect- 
angle is  equal  to  the  product  of  its 
base   and    altitude.     Then,    in    the 
rectangle  ABCD,  we  have 

Area  =  AB  x  EC. 

Take  a    second  rectangle   EFGH,  having  a     ggcond 
longer  base  EF,  and  a  less  altitude  FG,  but  such    J^=''™pio> 
that  it  shall   have   an   equal     h  g 

area  with  the  first :  then  we 
shall  have 


Area  =  EF  x  FG. 

Now  since  the  areas  are  equal,  we  shall  have 

AB  X  EC  =  EF  X  FG  ;  Equaiioa. 

and  by  resolving  the  terms  of  this  equation  into 
a  proportion,  we  shall  have 

AB  :  EF  :  :  FG  :  EC.  Proporuob 

It  is  plain  that  the  sides  of  the  rectangle  ABCD 
may  be  so  changed  in  value  as  to  become  the 
sides  of  the  rectangle  EFGH,  and  that  while 
they  are  undergoing  this  change,  AB  will  in- 
crease and  EC  diminish.      The  change  in  the  Relation?  oi 

the  quanti- 

values  of  these  quantities  will  therefore  take  place       ties.- 
according  to  a  fixed  law :  that  is,  one  will  be  di- 
minished as  many  times  as  the  other  is  increased, 


346  MATHEMATICAL     SCIENCE,  [bOOK  II. 

since   their  product  is  constantly  equal  to  the 
ai'ea  of  the  rectangle  EFGH. 
Expressed  by      Denote  the  side  AB  by  x  and  BC  by  y,  and 

letters. 

the  area  of  the  rectangle  EFGH,  which  is  known, 
by  a  ;  then 

xr/  =  a  ; 

and  when  the  product  of  two  varying  quantities 

is  constantly  equal  to  a  known  quantity,  the  two 

Reciprocal    quantities  are  said  to  be  Reciprocally  or  laverse- 

inverse  Pro-  ^  proportional ;  thus  X  and  y  are  said  to  be  in- 

poruon.     versely  proportional  to  each  other.     If  we  divide 

1   by  each  member  of  the  above  equation,  we 

shall  have 

J__l 
xy      a' 

Reductions   jmd  by  multiplying  both  members  by  x,  we  shall 

of  the 
Equations.     haVG 

}__X 

y~  a' 

and  then  by  dividing  both  numbers  by  x,  we  have 


final  form. 


1 

~      a 

X 


that  is,  the  ratio  of  a:  to  -  is  constantly  equal  to  -: 
y  ^    ^  a 

that  is,  equal  to  the  same  quantity,  ho^^ever  x  or 


CHAP.  III.] 


GEOMETRY, 


247 


y  may  vary ,  for,  a  and  consequently  -  does  not 

change.     Hence, 

Two  quantities,  which  may  change  their  values, 
are  reciprocally  or  inversely  proportional,  when 
one  is  proportional  to  unity  divided  by  the  other, 
and  then  their  product  remains  constant. 

We  express  this  reciprocal  or  inverse  relation 
thus : 

A  is  said  to  be  inversely  proportional  to  B  :  the 
symbols  also  express  that   A  is  directly  propor- 
tional to  -rr.     If  we  have 
Jj 

Aoc|, 

we  say,  that  A  is  directly  proportional  to  B,  and 
inversely  proportional  to  C. 

The  terms  Direct,  Inverse  or  Reciprocal,  ap- 
ply to 'the  nature  of  the  proportion,  and  not  to 
the  Ratio,  which  is  always  a  mere  quotient  and 
the  measure  of  proportion.  The  term  Direct  ap- 
plies to  all  proportions  in  which  the  terms  in- 
crease or  decrease  together ;  and  the  term  In- 
verse or  Reciprocal  to  those  in  which  one  term 
increases  as  the  other  decreases.  They  cannot, 
therefore,  properly  be  applied  to  ratio  without 
changing  entirely  its  signification  and  definition. 


Inverse 

Proporticm 

defined. 


How  ex- 
pressed. 


Generally, 
how  read. 


Direct  and 

Inverse, 

terms  not 

applicable  tc 

Ratio. 


248 


MATHEMATICAL     SCIENCE.  [bOOK  II. 


COMPARISON     OF     FIGURES, 


Geometrical 
magnitudes 
compared. 


Example. 


Formula  of 
Oomparisoii. 


Changes  of 

value: 
bow  affected 


Circles  com- 
parud. 


§  274.  In  comparing  geometrical  magnitudes, 
by  means  of  their  quotient,  it  is  not  the  quotient 
alone  which  we  consider.  The  comparison  im- 
plies a  general  relation  of  the  magnitudes,  which 
is  measured  by  the  Ratio.  For  example :  we 
say  that  "Similar  triangles  are  to  each  other  as 
the  squares  of  their  homologous  sides."  What 
do  we  mean  by  that  ?     Just  this  : 

That  the  area  of  a  triangle 

Is  to  the  area  of  a  similar  triangle 

As  the  area  of  a  square  described  on  a  side  ol 
the  first, 

To  the  area  of  a  square  described  on  an  ho- 
mologous side  of  the  second. 

Thus,  we  see  that  every  term  of  such  a  pro- 
portion is  in  fact  a  surface,  and  that  the  area 
of  a  triangle  increases  or  decreases  much  faster 
than  its  sides  ;  that  is,  if  vv^e  double  each  side  ol 
a  triangle,  the  area  will  be  four  times  as  great: 
if  we  multiply  each  side  by  three,  the  area  will 
be  nine  times  as  great ;  or  if  we  divide  each 
side  by  two,  we  diminish  the  area  four  times,  and 
so  on.     Again, 

The  area  of  one  circle 

Is  to  the  area  of  another  circle, 

As  a  square  described  on  the  diameter  of  the  first 


CHAP,  iir.] 


GEOMETRY, 


249 


To  a  squar-*  described  on  the  diameter  of  the 

second. 
Hence,  if  we  double  the  diameter  of  a  circle,    how  their 

ai'eas  change, 

the  area  of  the  circle  whose  diameter  is  so  in- 
creased will  be  fom'  times  as  great :  if  we  mul- 
tiply the  diameter  by  three,  the  area  will  be  nine     ^'"'='1''*' 

'■    ''  •'  general 

times  as  great ;  and  similarly  if  we  divide  the 
diameter. 


§  275.   In    comparing    volumes    together,    the  Comparison 
same  general  principles  obtain.     Similar  volumes 
are  to  each  other  as  the  cubes  described  on  their 
homologous  or  correspnuding  sides.     That  is, 

A  prism 

Formula. 

Is  to  a  similar  prism. 

As  a  cube  described  on  a  'ide  of  the  first, 

Is  to  a  cube  described  on  an  homologous  side 

of  the  second. 
Hence,  if  the  sides  of  a  prism  be  doubled,  the  con-     How  the 

V(>liime8 

tents  of  volume  will  be  increased  eight-fold.  Again,      change. 
A  sphere 

Is  to  a  sphere.  Sphere: 

As  a  cube  described  on  the  diameter  of  the  first. 
Is  to  a  cube  described   on  a  diameter  of  the 

second. 
Hence,  if  the  diameter  of  a  sphere  be  doubled,     now  it? 

vohime 

its  contents  of  volume  will  be  increased  eight-    changes, 
fold ;  if  the  diameter  be  multiplied  by  three,  the 


250  MATHEMATICAL     SCIENCE.  [BOOK   II. 

contents  of  volume  will  be  increased  twenty-seven 
fold :  if  the  diameter  be  multiplied  by  four,  the 
contents  of  volume  will  be  increased  sixty-fonr 
fold ;  the  contents  of  volume  increasing  as  the 
cubes  of  the  numbers  1,  2,  3,  4,  &c. 

Ratio:  §276.    The  relation  or  ratio  of  two  magnitudes 

an  abstract  to  eacli  Other,  maybe,  and  indeed  is,  expressed 

num  ei.  ^^  ^^^  abstract  number.  This  number  has  a 
whenhav-  fixed  valuc   SO   long   as  we   do  not   introduce  a 

ing  a  fixed 

value.  change  in  the  contents  of  the  figures;  but  if 
Ave  wish  to  express  their  ratio  under  the  sup- 
position that  their  contents  may  change  accord- 
ing to  fixed  laws  (that  is,  so  that  the  volumes 
How  varying  shall  coutiuue  similar),  we  then  compare  them 
compared.  With  similar  figures  described  on  their  homol- 
ogous or  corresponding  sides;  or,  what  is  the 
same  thing,  take  into  account  the  corresponding 
changes  which  take  place  in  the  abstract  num- 
bers that  express  their  contents. 


EEC APITULATIOK. 

General  §  277.  We  have  now  completed  a  general  out- 
line of  the  science  of  Geometry,  and  what  has 
been  said  may  be  recapitulated  under  the  follow- 
ing heads.     It  has  been  shown, 

Geometry:       1st.  That  Geometry  is  conversant  about  space. 


CHAP.   III.]  GEOMETET.  251 

or   those   limited    portions    of    space    wLicli   are    to  what  it 

relates. 

called  Geometrical  Magnitudes. 

2d.  That  the  geometrical  magnitudes  embrace 
four  species : 

1st.  Lines — straight  and  curved ;  Lines. 

2d.  Surfaces — plane  and  curved  ;  Surfaces. 

3d.  Volumes — bouiided  either  by  plane  sur-    volumes. 
faces  or  curved,  or  both  ;  and, 

4th.  Angles,  arising  from  the  positions  of     Angles. 
lines  with  each  other;  or  of  planes  with  each 
other — the  lines  and  planes  being  boundaries. 
3d.   That  the  science  of  Geometry  is  made  up     science: 
of  those  processes  by   means   of  which   all    the     "^p"'** 
properties  of  these  magnitudes  are  examined  and 
developed,  and  that  the  results  arrived  at  con- 
stitute the  truths  of  Geometry. 

4th.  That  the  truths  of  Geometry  may  be  di-      Truths 
vided  into  three  classes  :  three  classes. 

1st.  Those   implied  in  the  definitions,  viz.    Fiist  class. 
that    things    exist    corresponding   to    certain 
words  defined  ; 

2d.    Intuitive    or    self-evident    truths    em-      gecond. 
bodied  in  the  axioms  ; 

3d.  Truths  deduced  (that  is,  inferred)  from      ^^^.^j 
the  definitions  and  axioms,  called  Demonstra- 
tive Truths. 
5th.  That  the  examination  of  the  properties  of  Geometrical 
tlie  geometrical  magnitudes  has  reference,  magmtudoa. 


253 


MATHEMATICAL     SCIENCE, 


[book  II. 


Comparison. 


Properties. 


Proportion. 


1st.  To  their  comparison  with  a  standard 
or  unit  of  measure  ; 

2d.  To  the  discovery  of  properties  belong- 
ino;  to  an  individual  figure,  and  vet  common  to 
the  entire  class  to  which  such  figure  belongs ; 

3d.  To  the  comparison,  with  respect  to  mag- 
nitude, of  figures  of  the  same  species  with  each 
other;  viz.  lines  with  lines,  surfaces  with  sur- 
faces, volumes  with  volumes,  and  angles  with 
anffles. 


SUGGESTIOXS  FOR  THOSE  WHO  TEACH  GEOMETRY. 

Snggestiong.      1.  Be  sui'c  that  your  pupils  have  a  clear  ap- 
First.      prehension  of  space,  and  of  the  notion  that  Ge- 
1      ometry  is  conversant  about  space  only. 

2.  Be  sure  that  they  understand  the  significa- 
Second.     tioii  of  the  terms,  lines,  surfaces,  and  volumes, 

and  that  these  names  indicate  certain  portions 
of  space  corresponding  to  them. 

3.  See   that   they   understand    the   distinction 
Third,      between  a  straight  line  and  a  curve ;  between  a 

jolane  surface  and  a  curved  surface;  between  a 
volume  bounded  by  planes  and  a  volume  bounded 
by  curved  surfaces. 

4.  Be  careful  to  have  them  note  the  charac- 
•  Fourth,     teristics  of  the  different  species  of  plane  figures, 

such  as  triangles,  quadrilaterals,  pentagons,  hexa- 
gons, &c. ;   and  then  the  characteristic  of  each 


CHAP.   III.] 


G  E  0  31  E  T  R  Y . 


253 


class  or  subspecies,  so  tliat  the  name  shall  recall, 
at  once,  the  characteristic  properties  of  each 
figure. 

5.  Be   careful,    also,   to   have    them    note   the 
characteristic   diiferences   of  the   volumes.      Let      Fifths 
them   often   name   and   distinguish  those  which 

are  bounded  by  planes,  those  bounded  by  plane 
and  curved  surfaces,  and  those  bounded  by 
curved  surfaces  only.  Regarding  Volume  as  a 
genus,  let  them  give  the  species  and  subspecies 
into  which  it  may  be  divided. 

6.  Having  thus  made  them   familiar  with  the 
things  which  are  the  subjects  of  the  reasoning,      sixth, 
explain   carefully  the   nature  of  the  definitions; 

then  of  the  axioms,  the  grounds  of  our  belief  in 
them,  and  the  information  from  which  those 
self-evident  truths  are  inferred. 

7.  Then  explain  to  them,  that  the  definitions 

and  axioms  are  the  basis  of  all  geometrical  rea-     seventh, 
soning :  that  every  proposition  must  be  deduced : 
from  them,  and   that  they  afford  the  tests  of  all 
the  truths  w.hich  the  reasonings  establish. 

8.  Let  every  figure,  used  in  a  demonsiraticHi, 

be  accurately  drawn,  by  the  pupil  himself,  on  a      Eightb. 
blackboard.      This   will  establish    a    connection 
between  the  eye  and  the  hand,  and  give,  at  the 
same  time,  a  clear  perception  of  the  figure  and  a 
distinct  apprehension  of  the  relations  of  its  parts. 


254 


MATHEMATICAL     SCIENCE, 


[book  II, 


9.  Let  the  pupil,  in  every  demonstration,  first 
Ninth.      enunciate,  in   general  terms,  that  is,  \vithout  the 

aid  of  a  diagram,  or  any  reference  to  one,  tho 
proposition  to  be  proved ;  and  then  state  the 
principles  previously  established,  which  are  to 
be  employed  in  making  out  the  proof 

10.  When  in  the  course  of  a  demonstration, 
Tenth,      any  truth  is  inferred   from   its  connection  with 

one  before  known,  let  the  truth  so  referred  to  be 
fully  and  accurately  stated,  even  though  the 
number  of  the  proposition  in  which  it  is  proved, 
be  also  required.     This  is  deemed  important. 

11.  Let  the  pupil  be  made  to  understand  that 
Eleventh,  a  demonstration  is  but  a  series  of  logical  argu- 
ments arising  from  comparison,  and  that  the 
result  of  every  comparison,  in  respect  to  quan- 
tity, contains  the  mark  either  of  equality  or 
inequality. 

12.  Let    the    distinction    between    a    positive 
Twelfth,     and  negative  demonstration   be   fully  explained 

and  clearly  apprehended. 

13.  In  the  comparison  of  quantities  with  each 
Thirfeenth.  Other,  great  care  should  be  taken  to  impress  the 

fact  that  proportion  exists  only  between  quan- 
tities of  the  same  kind,  and  that  ratio  is  the 
measure  of  proportion. 

14.  Do  not  fail  to  give  much  importance  to 
Fourteenth,  the  kind  of  quantity  under  consideration.     Let 


CHAP.   III.]  GEOMETEY.  255 

the  question  be  often  put,  What  kind  of  quantity  Fourteenth, 
are  3'ou  considering  ?     Is  it  a  line,  a  surface,  or 
a  volume  ?     And  what  kind  of  a  line,  surface,  or 
volume  ? 

15.  In  all  cases  of  measurement,  the  unit  of 
measure    should   receive    special    attention.       If 

lines  are  measured,  or  compared  by  means  of  a  Fifteenth, 
common  unit,  see  that  the  pupil  perceives  that 
unit  clearly,  and  apprehends  distinctly  its  rela- 
tions to  the  lines  which  it  measures.  In  sur- 
faces, take  much  pains  to  mark  out  on  the 
blackboard  the  particular  square  which  forms 
the  unit  of  measure,  and  wTite  unit,  or  unit  of 
measure,  over  it.  So  in  the  measurement  of 
volumes,  let  the  unit  or  measuring  cube  be  ex- 
hibited, and  the  conception  of  its  size  clearly 
formed  in  the  mind  ;  and  then  impress  the  im- 
portant fact,  that,  all  measurement  consists  in 
merely  comparing  a  unit  of  measure  with  the 
quantity  measured ;  and  that  the  number  which 
expresses  the  ratio  is  the  numerical  expression 
for  that  measure. 

16.  Be  careful  to  explain  the  difference  of  the   sixteenth, 
terms  Equal  and  Equal  in  all  their  parts,  and 

never  permit  the  pupil  to  use  the  terms  as  sy- 
nonymous. An  accurate  use  of  Avords  leads  to 
nice  discriminations  of  thought. 


CHAP.  IV  ] 


AN  ALY  SIS. 


257 


CHAPTER   IV. 

ANALYSIS— ALGEBRA  —  ANALYTICAL     GBOaiETBT. 

ANALYSIS. 

§  278.    Analysis    is    a   general  term,    embra-     Analysis 
cing  that  entire  portion  of  mathematical  science 
in  which  the  quantities   considered    are   repre- 
sented by  letters  of  the  alphabet,  and  the  opera- 
tions to  be  performed  on   them  are  indicated  by 


§  279.  We  have  seen  that  all  numbers  must    Numbers 

must  be  of 

be  numbers  of  something  ;*  for,  there  is  no  such      things; 
thing  as  a  number  without  a  basis  :  that  is,  every 
number  must  be  based  on  the  abstract  unit  one, 
or  on   some  unit    denominated.       But    although 
numbers  must  be  numbers  o^  something,  yet  they  butmaybe 

.  of  many  kind 

may  be  numbers  of  anij  thing,  for  the  unit  may    of  things. 
be  whatever  we  choose  to  make  it. 


*  Section  112. 
17 


2j:5  mathematical    SCIEisGE.  fBOOK   11, 


All  quantity        §  280.    All  quantity,  made  up  of  definite  parts, 

consists  of  ,  ,  ,  ,  .  .       , 

parts.       can   be    numbered   exactly   or    approximatively, 

and,  in  this  respect,  possesses  all  the   properties 

of  number.     Propositions,   therefore,  concerning 

numbers,  have  the  remarkable   peculiarity,   that 

Propositions  they   are  propositions  concerning  all  quantities 

in  regard  to 

number     whatevcr.     That  half  of  six  is  three,  is  equally 

jpply  also  to  ,  ,  ,         . 

quantity,  true,  Whatever  the  word  six  may  represent, 
whether  six  abstract  units,  six  men,  or  six  tri- 
angles. Analysis  extends  the  generalization  still 
further.  A  number  represents,  or  stands  for,  that 
particular  number  of  things  of  the   same  kind, 

Algebraic     without  reference   to  the   nature  of  the  thing ; 

symbols 

gener-  but  an  analytical  symbol  does  more,  for  it  may 

stand  for  all  numbers,  or  for  all  quantities  which 

numbers  represent,  or  even  for  quantities  which 

cannot  be  exactly  expressed  numerically. 

Any  thing        As  soou  as  wc  coucclve  of  a  thing  we  may 

maybedi-   couceivc   it  divided  into  equal  parts,  and  may 

^"*'''''       represent  either  or  all  of  those  parts  by  a  or  x, 


more 
al. 


or  may,  if  we  please,  denote  the  thing  itself  by  a 
or  X,  without  any  reference  to  its  being  divided 
into  parts. 

Each  figure       §  281.    Li  Gcometfy,  each  geometrical  figure 

stands  for  a  i       r-  i  i        i  i  i 

.'lass       stands  lor  a  class ;  and  when  we  have  demon, 
strated  a  property  of  a  figure,  that  property  ' 
considered  as  proved  for  every  figure  of  the  class. 


CHAP.   IV.]  AN-ALYSIS,  259 

For  example:   when  we   prove   that   the   square    Example, 
described  on  the  hypothenuse  of  a  right-angled 
triangle  is  equal  to  the  sum  of  the  squares  de- 
scribed on  the  other  two  sides,  we  demonstrate  the 
fact  for  all  right-angled  triangles.     But  in  analy-  in  analysis 
sis,  all   numbers,  all   lines,  all   surfaces,  all  vol-    stand  foi- 
umes,  may  be  denoted  by  a  single  symbol,  a  or  x.     ciass^-s. 
Hence,   all   truths   inferred   by   means   of   these 
symbols  are  true  of  all  things  whatever,  and  not 
like   those   of  number  and  geometry,  true    only 
of  particular  classes  of  things.     It  is,  therefore, 
not  surprising,  that   the  symbols  of  analysis  do 
not  excite  in  our  minds  the  ideas  of  particular    Hence,  i&e 

,  .  mi        ■  •  I  J  ?     truths  inf'ef- 

things.      1  he  mere  written  characters,  a,  o,  c,  a,  red  are  gei> 
X,  y,  z,  serve  as  the  representatives  of  things  in 
general,  whether  abstract  or  concrete,  whether 
known  or  unknown,  whether  finite  or  infinite. 


§  282.   In   the   uses  which  we  make  of  these     Symbols 

come  to  be 

symbols,  and  the   processes  of  reasoning  carried    regarded  m 
on  by  means  of  them,  the  mind  insensibly  comes 
to  regard  them  as  things,  and  not  as  mere  signs ; 
and  we   constantly  predicate  of  them   the  prop> 
erties  of  things  in  general,  without   pausing  to 
inquire   what  kind  of  thing  is   implied.      Thus,     Example, 
we   define   an  equation   to   be   a  proposition    in    xheequ* 
which  equality   is    predicated   of   one  thing    as       ^'°"" 
compared  with  another.     For  example  : 


What  axioms 


260  MATHEMATICAL     SCIENCE.  ("bOOK   II 


a  -^  C  =  X, 

is    an    equation,    because  x   is    declared    to    be 
necessai-y to  gg^j^l  to  the  sum  of  a  and  c.     In  the  solution  ol 

its  solution.        • 

equations,  we  employ  the   axioms,  "  If  'equals  be 

added  to  equals,  the  sums  will  be  equal ;"  and, 

"  If  equals  be  taken  from  equals,  the  remainders 

They  express  will  be  equal."     Now,  these  axioms  do  not  ex- 

qualities  of  ^  r     ^  •  c 

things,      press    qualities   of    language,   but    properties    of 
Hence,  in-    quantity.       Hence,    all  inferences  in  mathemat- 

ferencesre-     .  .  i     i  i     i  i       i       •  ^• 

late  to  things,  ical  sciencc,  deduced  through  the  instrumentality 

of  symbols,   whether  Arithmetical,  Geometrical, 

or  Analytical,  must  be  regarded  as  concerning 

quantity,  and  not  symbols. 

Quantity         As  analytical  symbols  are  the  representatives 

nee  not  a-   ^  quantity  in  seneral,  there  is  no  necessity  of 

waysbepres-  1  J  o  '  J 

enttothe    ]^eeping  the  idea  of  quantity  continually  alive  in 

mind. 

the  mind ;    and  the  processes  of  thought  may, 
without  danger,  be  allowed  to  rest  on  the  sym- 
bols themselves,  and  therefore,  become   to  that 
extent,  merely  mechanical.     But,  when  we  look 
The  reason-  back  and  scc  ou  what  the  reasoning  is  based,  and 
based  on'' the  ^ow  the  proccsscs  have  been  conducted,  we  shall 
supposition   ^j^^  ^]^jj|,  gygj-y  gtgp  ^as  taken  on  the  supposition 

of  quantity.  ^  r  i  i 

that  we  were  actually  dealing  with  things,  and 
not  symbols ;  and  that,  without  this  understand- 
ing of  the  language,  the  whole  system  is  without 
signification,  and  fails. 


CHAP.   IV,]  A^STALYSIS.  261 


§  283.  There  are  ihree  principal  brar.clies  of     Three 

.        ,       .  branches    ^ 

Analysis  : 

1st.    Algebra.  Algebra,    ■ 

2d.  Analytical  Geometry.  Anaiyticaj 

"  Geometry, 

3d.  Differential  and  Integral  Calculus.  caicute. 


ALGEBRA. 

§  284.  Algebra  is,  in  fact,   a  species  of  uni-     Algebra: 
versal  Arithmetic,  in  which  letters  and  signs  are     universal 
employed  to  abridge  and  generalize  all  processes     ^'  '"^"*' 
involving  numbers.     It  is  divided  into  two  parts,    Two  parts 
corresponding  to  the   science  and   art  of  Arith- 
metic : 

1st.  That  which  has  for  its  object  the  investi-    First  part. 
gation  of  the   properties  of  numbers,  embracing 
all    the  processes    of  reasoning  by  which    new 
properties  are  inferred  from  known  ones  ;  and, 

2d.  The   solution  of  all  problems  or  questions  second  pan. 
involving  the  determination  of  certain  numbers 
which  are  unknown,  from  their  connection  with 
certain  others  which  are  known  or  given. 


ANALYTICAL     GEOMETKY, 


§  285.    Analytical    Geometiy    examines    the  Analytical 

,  ,      .  r      1  Geometry. 

pi'operties,   measures,   and  relations   oi   the  geo- 
metrical magnitudes  by  means  of  the  analytical  itmature. 


262  MATHEMATICAL     SCIENCE.  [bOCK   II. 


symbols.     This  branch  of  mathematical  science 
oescartes,     originated  with  the  illustrious  Descartes,  a  cele- 

the  original 

founder  of   bratcd  Frcnch  mathematician  of  the  17th  cen- 
tury.    He  observed  that  the  positions  of  points, 

What  he  *' 

observed,    the  direction  of  lines,  and  the  forms  of  surfaces, 

could  be  expressed  by   means  of  the   algebraic 

Au  position  symbols ;  and  consequently,  that  every  change, 

expressed  by  .  .   .  . 

symbols.  Cither  HI  the  position  or  extent  oi  a  geometrical 
magnitude,  produced  a  corresponding  change  in 
certain  symbols,  by  which  such  magnitude  could 
be  represented.  As  soon  as  it  was  found  that, 
to  every  variety  of  position  in  points,  direction 
in  lines,  or  form  of  curves  or  surfaces,  there  cor- 
responded certain  analytical  expressions  (called 
their  Equations),  it  followed,  that  if  the  processes 
were  known  by  which  these  equations  could  be 
The  equation  examined,  the  relation  of  their  parts  determined, 

develops  the  i  •    i         i 

properties    and   thc  laws    according   to   which   those    parts 
''niiude'^'  Vary,  relative  to  one  another,  ascertained,  then 
the    corresponding  changes    in    the  geometrical 
magnitudes,   thus  represented,    could    be    imme- 
diately inferred. 

Hence,  it  follows  that  every  geometrical  ques- 
Powerover   tion  Can  be  solved,  if  we  can  resolve  the  corre- 

the  magni-  i       i        •  •  i     i 

tilde  extend-  spondmg  algcbraic  equation  ;  and  the  power  over 
edbythe    ^j^^  freometrical  magnitudes  was  extended  just  in 

equatton.  &  o  j 

proportion  as   the  properties    of   quantity  were 
brought  to  light  by  means  of  the  Calculus.     The 


CHAP.  IV.']  ANALYSIS.  263 

applications  of  this  Calculus  were  soon  made  to  to  what  sub- 

ject  applied. 

the   subjects   of  mechanics,   astronomy,   and  m- 

deed,  in  a  greater  or  less  degree,  to  all  branches 

of  natural    philosophy ;    so  that,  at  the  present    its  present 

time,   all    the   varieties   of  physical  phenomena, 

with  which  the  higher  branches  of  the  science 

are  conversant,  are  found  to  answer  to  varieties 

determinable  by  the  algebraic  analysis. 


8  286.  Two  classes  of  quantities,  and  conse-    Quantities 

which  enter 

quently  two  sets  of  symbols,  quite  distinct  from  into  the  gu 

cuius. 

each  other,  enter  into  this  Calculus ;    the   one 

called  Constants,  which  preserve  a  fixed  or  given    constants. 

value  throughout  the  same  discussion  or  investi- 

gation  ;    and   the  other  called   Variables,  which    variables. 

undergo   certain   changes   of  value,   the  laws  of 

which  are  indicated  by  the  algebraic  expressions 

or  equations  into  which  they  enter.     Hence, 

Analytical  Geometry  may  be  defined  as  that    Analytical 

Geometry 

branch  of  mathematical  science,  which  exam-  defined.' 
ines,  discusses,  and  develops  the  properties  of 
geometrical  magnitudes  by  noting  the  changes 
that  take  place  in  the  algebraic  symbols  which 
represent  them,  the  laws  of  change  being  deter- 
mined by  an  algebraic  equation  or  formula. 


264' 


MATHEMATICAL     SCIENCE, 


[book  IL 


DIFFERENTIAL  AND  INTEGRAL,  CALCULUS. 

Quantities        g  287    Jn  this  branch  of  mathematical  science. 

considered. 

as  in  Analytical  Geometry,  two  kinds  of  quan- 
variabies,    ^j^y  ^^.g  considered,  viz.  Variables  and  Constants  ; 

Constants. 

and   consequently,  two  distinct  sets  of  symbols 
The  Science,  are  employed.     The  science  consists  of  a  series 
of  processes   which  note  the  changes  that  take 
place   in    the    value   of    the    Variables.       Those 
changes  of  value  take  place  according  to  fixed 
laws  established  by  algebraic  formulas,  and  are 
Marks,      indicated  by  certain  marks  drawn  from  the  va- 
Differen-    riablc   symbols,    called   Differentials.      By   these 
marks   we   are    enabled   to   trace   out   with   the 
accuracy  of  e.xact  science  the  most  hidden  prop- 
erties of  quantity,  as  well  as  the  most  general 
and  minute  laws  Avhich  regulate  its  changes  of 
value. 


Analytical         §  288.     It  will    be    obscrved,   that  Analytical 

ami   '    Geometry  and  the  Differential  and  Integral  Cal- 

Caicuius:    p^jyg  treat  of  quantity  regarded  under  the  same 

general  aspect,  viz.  as  subject  to  changes  or  va- 

Howtiey    riations  in  magnitude  according  to  laws  indica- 

regard  qiian- 

tity:       ted  by  algebraical  formulas;   and  the  quantities, 

whether  variable  or  constant,  are,  in  both  cases, 

by  what     represented  by  the  same  algebraic  symbols,  viz. 

represente  .  ^j^^  constants  by  the  first,  and  the  variables  by 


CHAP.   IV.]  ALGEBRA.  265 

the  find  letters  of  the  alphabet.     There  is,  how-    Difference; 
ever,    this    important    difference :    in   Analytical 
Geometry  all  the  results  are   inferred  from  the    in  what  it 

consists, 

lelations  which  exist  between  the  quantities 
themselves,  while  in  the  Differential  and  Integral 
Calculus  they  are  deduced  by  considering  what 
may  be  indicated  by  marks  drawn  from  variable 
quantities,  under  certain  suppositions,  and  by 
marks  of  such  marks. 


§  289.  Algebra,  Analytical  Geometry,  the  Dif-    Analytical 

f~\  1  Science. 

ferential  and  Integral  Calculus,  extended  into  the 
Theoi'y  of  Variations,  make  up  the  subject  of 
analytical  science,  of  which  Algebra  is  the  ele- 
mentary branch.  We  shall,  in  this  chapter, 
limit  our  remarks  to  the  subject  of  Algebra ;  Algebra, 
reserving  a  separate  chapter  for  the  Differential  Differential 

Calculus. 

and  Integral  Calculus.     This  subject  embraces  a 
very  remarkable  class  of  quantities. 


ALGEBRA. 

§  '290.  On  an  analysis  of  the  subject  of  Alge-  Algebra, 
bra,  we  think  it  will  appear  that  the  subject  itself 

presents  no   serious  difficulties,  and  that  most  of  Difficuitiea 
the  embarrassment  which  is  experienced  by  the 

pupil  in  gaining  a  knowledge  of  its  principles,  as  However 

well  as  in  their  applications,  arises  from  not  at  '^^^^ 


2GG  MATHEMATICAL     SCIENCE.  [bOOK  II. 

Language,  tending  sufficiently  to  the  language  or  signs  of 
the  thoughts  which  are  combined  in  the  reason- 
ings. At  the  hazard,  therefore,  of  being  a  httle 
diffuse,  I  shall  begin  with  the  very  elements  of 
the  algebraic  language,  and  explain,  with  much 
minuteness,  the  exact  signification  of  the  char- 

characters    actcrs  that  Stand  for  the  quantities  which  are  the 

which  repre-         ,  .  „     ,  ,       .  ,       ,  ^     , 

Bentquantity.  suDjccts  ot  the  aualysis ;  and  also  oi  those  signs 
Signs.       which  indicate  the  several  operations  to  be  per- 
formed on  the  quantities. 

Quantities.         §  291.  The  quantities  which  are  the  subjects 

Howdivided.  of  the    algebraic    analysis  may  be  divided  into 

two  classes :  those  which  are  known  or  given, 

and  those  which  are  unknown  or  souo-ht.     The 

o 

How  repre-  knowu  are  uniformly  represented  by  the  first 
letters  of  the  alphabet,  a,  h,  c,  d,  &c. ;  and  the 
unknown  by  the  final  letters,  x,  y,  z,  v,  w,  &c. 


May  be  in- 
creased or 


§  292.     Quantity  is    susceptible  of  being  in- 
creased or  diminished  ;     and  there  are   six   oper- 

itiminished. 

Five  opera-  atious  which  Can  be  performed  upon  a  quantity 
''°"*'       that  will  give  results  differing  from  the  quantity 

it&^lf,  viz. : 
Piret,  ^^^-  ^^  ^^^  ^^  ^^  itself  or  to  some  other  quan- 

tity; 
Second.         2d.  To  Subtract  some  other  quantity  from  it; 


CHAP.   IV  J  ALGEBRA  26? 


3d.  To  multiply  it  by  a  number ;  Third. 

4tli.   To  divide  it;  Fourth. 

5tli.  To  raise  it  to  any  power;  and  ^if^^- 

6th.    To  extract  a  root  of  it.  sixth. 

The  cases  in  which  the  multiplier  or  divisor 

is  1,   are  of  course   excepted ;   as   also    the  case  Exception, 
where  a  root  is  to  be  extracted  of  1. 

§  293.  The  six  signs  which  denote  these  oper-  signs. 
ations  are  too  well  known  to  be  repeated  here. 

These,  with  the  signs  of  equality  and  inequality,  Elements 

of  the 

the  letters  of  the  alphabet  and  the  figures  which  Algebraic 

are  employed,  make  up  the  elements  of  the  alge-  "^^^^^^ 

braic  language.     The  words   and  phrases   of  the  its  words 

algebraic,  like  those  of  every  other  language,  are  '   i"™^'''' 
to  be  taken  in  connection  with  each  other,  and 

are  not  to  be  interpreted  as  separate  and  isolated  How  inter 

preted. 

.symbols. 

§  29-4.  The  symbols  of  quantity  are  designed  symbols  of 

to  represent  quantity  in  general,  whether  abstract  *i"'^'''y- 
or   concrete,  whether  known  or  unknown ;  and 

the   signs  which  indicate   the   operations   to    be  General. 
performed  on  the  quantities  are  to  be  interpreted 
in  a  sense  equally  general.     When  the  sign  plus 

is  written,  it  indicates   that  the  quantity  before  Examples. 

which  it  is  placed  is  to  be  added  to  some  other  signs  plus 

quantity ;  and  the  sign  minus  implies  the  exist-  °^  """"* 


268  MATHEMATICAL     SCIENCE.  [BOOK  II. 

ence  of  a  minuend,  from  which  the  subtrahend 

is  to  be  taken.     One  thing  should  be  observed  in 

Signs  havo   regard  to  the  signs  which  indicate  the  operations 

no  effect  on 

the  nature  of  that  are  to  be  performed  on  quantities,  viz.  they 

a  quantity.  77        /r  77  />     j 

do  not  at  all  affect  or  change  the  nature  of  the 
quantity  before  or  after  ivhich  they  are  written 
hut  merely  indicate  lohat  is  to  he  done  with  the 

Examples:    quantity.      In  Algebra,  for  example,  the   minus 

n  Aige  ra.  ^j^^^  merely   indicates  that   the  quantity   before 

which    it   is  written    is    to    be    subtracted   from 

In  Analytical  somc  Other  quantity ;  and  in  Analytical  Geom- 

Geometry.  .  .  .       ^  .  . 

etry,  that  the  line  before  which  it  falls  is  esti- 
mated in  a  contrary  direction  from  that  in  which 
it  would  have  been  reckoned,  had  it  had  the  sign 
plus  ;  but  in  neither  case  is  the  nature  of  the 
quantity  itself  different  from  what  it  would  have 
been  had  it  had  the  sign  plus. 

intcrpreta-        The  interpretation  of  the  language  of  Algebra 

language:    ^^  the  first  thing  to  which  the  attention  of  a  pupil 

should  be  directed  ;  and  he  should  be  drilled  on 

the   meaning   and  import   of  the  symbols,   until 

their  significations   and  uses  are  as  familiar  as 

Its  necessity,  the  souuds  and  combinations  of  the  letters  of  the 
alphabet. 

Elements         §  295.     Beginning  with  the   elements    of  the 
expamc  .    jj^j^g^ggg^  |g^  ^^^  number  or  quantity  be  desig- 
nated by  the  letter  a,  and  let  it  be  required  to 


CHAP.    IV. J 


ALGEBRA 


269 


add  this  letter  to  itself,  and  find  the  result  or  sum. 
The  addition  will  be  expressed  by 

a  -{-  a  =  the  sum. 

But  how  is  the  sum  to  be  expressed  ?     By  simply  signification 
regarding  a  as  one  a,  or  la,  and  then  observing 
that  one  a  and  one  a  make  two  a's  or  2  <2 :  hence, 

a  -\-a  =2a; 

and  thus  we  place  a  figure  before  a  letter  to  in- 
dicate how  many  times  it  is  taken.  Such  figure 
is  called  a  Coefficient.  Coefficieat. 


§  296.  The  product  of  several  numbers  is  in- 
dicated by  the  sign  of  multiplication,  or  by  sim- 
ply writing  the  letters  which  represent  the  num- 
bers by  the  side  of  each  other.     Thus, 

a  X  h  X  c  X  d  xf,  or  abcdf, 


Product; 


how  indies 
U?d 


indicates  the  continued  product  of  a,  b,  c,  d,  and  ' 
f,  and  each  letter  is  called  a  factor  of  the  prod- 
uct :  hence,  a  factor  of  a  product  is  one  of  the      Factor, 
multipliers  which  produce  it.     Any  figure,  as  5, 
written  before  a  product,  as 

5  abcdf, 

is  the  coefficient  of  the  product,  and  shows  that  coefBciont  ot 
the  product  is  taken  5  times.  ap,    uc 


270  MATHEMATICAL    SCIEXCE.  [BOOK   II. 

Equal  fac-       §  297.    If  in   tlic  product  ahcdf,  the  numbers 

tors : 

represented  by  a.  h,  c,  d,  and  /  were  equal  to  each 
product     other,  they  would  each  be  represented  by  a  single 

becomes. 

letter  a,  and  the  product  would  then  become 

a  X  aX  aX  aXa  =  a^ ; 

How       that  is,  we  indicate  the  product  of  several  equal 

expressc  .   £^^^^^,g  -^^   simply  Writing   the  letter   once   and 

placing  a  figure  above  and  a  little  at  the  right 

of  it,  to  indicate  how  many  times  it  is  taken  as 

Exponent:  a  factor.     The  figure  so  Avritteu  is  called  an  ex- 

whQTo  writ-  ponent.    Hence,  an  exponent  denotes  how  many 

equal  factors  are  employed.     The  result  of  the 

multiplications,  is  called  the  5th  Pozver  of  a. 

Division:  §  398.  The  division  of  one  quantity  by  an- 
how  other  is  indicated  by  simply  writing  the  divisoi 
below  the  dividend,  after  the  manner  of  a  frac- 
tion ;  by  placing  it  on  the  right  of  the  dividend 
with  a  horizontal  line  and  two  dots  between  them; 
or  by  placing  it  on  the  right  with  a  vertical  line 
between  them :  thus  either  form  of  expression, 

Three  ^         i,_^a,        or        d  I  a, 

forms.  flj 

indicates  the  division  of  b  by  a. 

Roots:         §  299.  The  extraction  of  a   root    is   indicated 
howincii-    \)j  the  Sign  a/.     This  sign,  when  used  by  itself 

cated. 

indicates   the  lowest  root,  viz.  the   square  root. 


CHAP.  IV.]  ALGEBRA.  271 

If  any  other  root  is  to  be  extracted,  as  the  third, 
fourth,  fifth,  &c.,  the  figure  marking  the  degree      index; 
of  the   root  is  written  above  and  at  the  left  of  where  writr 

ten, 

the  sign  ;  as, 

'V~  cube  root,  -^  fourth  root,  &c. 

The  figure  so  written,  is  called  the  Index  of  the 
root. 

We  have  thus  given  the  very  simple  and  gen-     Language 
erai   language  by  which  we  indicate  every  one    operations 
of  the   six  operations  that  may  be  performed  on 
an  algebraic  quantity,  and   every  process  in  Al- 
gebra involves  one  or  other  of  these  operations. 


JM  I  N  U  S     SIGN. 

§  300.  The  algebraic  symbols  are  divided  into    Algebraic 

language ! 

two  classes  entirely  distinct   from   each   other, 

viz.   the   letters   that  are  used   to  designate  the  iiow  divided. 

quantities  which  are  the  subjects  of  the  science, 

and  the  signs  which  are  employed  to  indicate 

certain    operations    to   be   performed    on    those 

quantities.     We  have  seen  that  all  the  algebraic     Algebraic 

processes: 

processes  are  comprised  under  addition,  subtrac- 

*  '  their  num- 

tion,  multiplication,  division,  and  the  extraction       ber. 
of  roots  ;  and  it   is  plain,   that   the   naUire  of  a      bo  not 

..  x^iii  11  <-•  •       change  the 

quantity  is  not  at  ail  changed  by  prefixing  to  it  „jit,ire  of  the 
the  sign  which  indicates  either  of  these  opera-    ''"''""''*'»' 


272  MATHEMATICAL     SCIENCE.  [bOOK  II. 

tions.  The  quantity  denoted  by  the  letter  a,  for 
example,  is  the  same,  in  every  respect,  whatever 
sign  maybe  prefixed  to  it;  that  is,  whether  it 
be  added  to  another  quantity,  subtracted  from 
it,  whether  multiplied  or  divided  by  any  number, 
or  whetlier  we  exti-act  the  square  or  cube  or  any 
Algebraic    other  root  of  it.     The  algebraic  signs,  therefore, 

signs: 

how  regard-  must  be  regarded  merely  as  indicating  opera- 
tions to  be  performed  on  quantity,  and  not  as 
afiecting  the  nature  of  the  quantities  to  which 
they  may  be  prefixed.  We  say,  indeed,  that 
piu3  and  quantities  are  plus  and  minus,  but  this  is  an  ab- 
Minus.  breviated  language  to  express  that  they  are  to 
be  added  or  subtracted. 


Principles  of      §  301.   In  Algebra,  as  in  Arithmetic  and  Ge- 
thescience:  Q,^-,g|-j,y^  ^^jj  j^j-jg  principles  of  the  scicucc  are  de- 


From  wliat 
QeJuccd. 


Example. 


What  we 
wish  to  dis- 
cover. 


duced  from  the  definitions  and  axioms  ;  and  the 
rules  for  performing  the  operations  are  but  di- 
rections framed  in  conformity  to  such  principles. 
Having,  for  example,  fi\;ed  by  definition,  the  power 
of  the  minus  sign,  viz.  that  any  quantity  before 
which  it  is  written,  shall  be  regarded  as  to  be 
subtracted  from  another  quantity,  we  wish  to 
discover  the  process  of  performing  that  subtrac- 
tion, so  as  to  deduce  therefrom  a  general  2)rin- 
ciple,  from  which  we  can  frame  a  rule  applic  able 
to  all  similar  cases. 


lAP.   IV.]  ALGEBRA.  273 


SUBTRACTION. 

§  302.  Let  it  be  required,  for  example,  to  subtraction. 
subtract  from  b  the  difference  be- 
tween a  and  c.  Now,  having  writ- 
ten the  letters,  with  their  proper 
signs,  the  language  of  Algebra  expresses  that  it 
is  the  difference  only  between  a  and  c,  which  is 
to  be  taken  from  b ;  and  if  this  difference  were  Difference, 
known,  we  could  make  the  subtraction  at  once. 
But  the  nature  and  generality  of  the  algebraic 
symbols,  enable  us  to  indicate  operations,  merely,    operations 

.  indicated. 

and  we  cannot  m  general  make  reductions  until 
we  come  to  the  final  result.  In  what  general 
way,  therefore,  can  we  indicate  the  true  differ- 
ence ? 

If  we  indicate  the  subtraction  of 
a  from  b,  we  have  b  —  a;  but  then 
we  have  taken  away  too  much  from 
b  by  the  number  of  units  in  c,  for  it  was  not  a, 
but  the  difference  between  a  and  c  that  was  to 
be  subtracted  from  b.  Having  taken  away  too 
much,  the  remainder  is  too  small  by  c :  hence, 
if  c  be  added,  the  true  remainder  will  be  express- 
ed by  6  —  a  -f  c. 

Now,  by  analyzing  this  result,  we  see  that  the   Analysis cr 

the  result. 

Sign  of  every  term  of  the  subtrahend  has  been 
changed ;    and  what  has   been  shown  with  re- 


b  —  a 

Final 

formula. 

b  —  a  +  c 

274 


MATHEMATICAL     SCIENCE, 


[book  II 


Generaiiza-  spcct  to  these  quantities  is  equally  true  of  all 
others  standing  in  the  same  relation :  hence,  we 
have  the  following  general  rule  for  the  subtrac- 
tion of  algebraic  quantities  : 

Change  the  sign  of  every  term  of  the  suhtra- 
^'^^-       hend,  or  conceive  it  to  be  changed,  and  then  unite 
the  quantities  as  in  addition. 


MULTIPLICATION 


Multiplica- 
liun. 


§  303.  Let  us  now  consider  the  case  of  mul- 
tiplication, and  let  it  be  required  to  multiply 
a  —  b  by  c.     The  algebraic  language  expresses 


Signification  that  the  difference  between  a  and  b 

of  the         . 

language,    is    to   be    taken    as   many  times    as 
there  are  units  in  c.     If  we  knew 


a  —  b 
c 


ac—bc 


Process : 


this  difference,  we  could  at  once 
perform  the  multiplication.  But  by  what  gen- 
eral process  is  it  to  be  performed  without  finding 
that  difference  ?  If  we  take  a,  c  limes,  the  prod- 
uct will  be  ac ;  but  as  it  was  only  the  difference 
between  a  and  b,  that  was  to  be  multiplied  by  c. 
Its  nature,  this  product  ac  will  be  too  great  by  b  taken  c 
times ;  that  is,  the  true  product  will  be  expressed 
by  ac  —  bc:  hence,  we  see,  that. 
Principle  for  If  CI  Guantitij  having  a  plus  sign  be  multi- 
plied by  another  quantity  having  also  a  phis 
sign,  the  sign  of  the  product  will  be  plus ;  and 


the  sigi\s. 


CHAP.  IV.]  /VLGEBRA.  275 

if  a  quantity  having  a  minus  sign  be  m,ulti- 
plied  by  a  quantity  having  a  plus  sign,  the  sign 
of  the  product  will  be  minus. 

§  304    Let   us  now  take   the    most   general  ceneraicase 
case,  viz.  that  in  which  it  is  required  to  multi- 
ply a  —  b  hy  c  —  d. 

Let  us  again  observe  that  the  ali^ebraic  Ian- 
guage  denotes  that  a  —  6  is 
to  be  taken  as  many  times 
as  there  arc  units  in  c—d; 
and  if  these  two  differences 
were  known,  their  product 


a  —  b 

C — d  Ifs  fon«. 


ac — be 

—ad-\~bd 


ac  —  bc^ad-V  bd 


would  at  once  form  the  product  required. 

First :  let  us  take  a  —  6  as  many  times  as  there   First  step, 
are  units   in  c  ;  this  product,  from  what  has  al- 
ready been   shown,   is   equal   to  ac  —  be.      But 
since  the  multiplier  is  not  c,  but  c  —  d,  it  follows 
that  this  product  is   too  large  by  a  —  &  taken  d 
times  ;  that  is,  by  ad  —  bd:  hence,  the  first  prod-  second  step 
uct  diminished   by  this  last,  will  give  the  true 
product.     But,  by  the   rule  for  subtraction,  this 
difference  is  found  by  changing  the  signs  of  the  Howtakeu, 
subtrahend,  and  then  uniting  all  the  terms  as  in 
addition :  hence,  the    true  product   is  expressed 
by  ac  —  be  —  ad -^  bd. 

By   analyzing  this    result,    and  employing  an    Anaij-sisof 
abbreviated  language,  we  have  the  following  gen- 


276 


MATHEMATICAI,     SCIENCE, 


[bock  II. 


eral  principle  to  which  the  signs  conform  in  mul- 
tiplication, viz. : 

Plus  multiplied  hy  jjIus  gives  plus  in  the  prod- 
uct ;  plus  multiplied  hy  minus  gives  minus  ;  mi- 
nus multiplied  hy  plus  gives  minus  ;  and  minus 
multiplied  by  minus  gives  plus  in  the  product. 


General 
Principle. 


Remark. 


Particular 
case. 


Minus  sign : 


§  305.  The  remark  is  often  made  by  pupils 
that  the  above  reasoning  appears  very  satisfac- 
tory so  long  as  the  quantities  are  presented  un- 
der the  above  form  ;  but  why  will  —h  multiplied 
by  —d  give  plus  hd  ?  How  can  the  product  of 
two  negative  quantities  standing  alone  be  plus  ir 
In  the  first  place,  the  minus  sign  being  pre- 
fixed to  b  and  d,  shows  that  in  an  algebraic  sens^ 
they  do  not  stand  by  themselves,  but  are  con- 
Jisinterpre-  ncctcd  with  Other  quantities;    and   if  they   are 

talion. 

not  so  connected,  the  minus  sign  makes  no  dif- 
ference ;  for,  it  in  no  case  affects  the  quantity, 
but  merely  points  out  a  connection  with  other 
quantities.       Besides,    the    product    determined 
above,  being  independent  of  any  particular  value 
attributed  to  the  letters  a,  h,  c,  and  d,  must  he 
Form  of  the  of  such  a  form  as  to  be  true  for  all  values ;  and 
must  be  true  heucc  for  the  case  in  which  a  and  c  are  both 
'or  quantities  g       I    ^^   ^ero.      Making    this    supposition,    the 

iif  any  value.      ^  o  i  i 

product  reduces  to  the  form  of  +  bd.     The  rule.'; 
for  the  signs  in  division  are  readily  deduced  from 


CUAP.   IV.]  ALGEBRA.  277 


the  definition  of  division,  and  the  principles  al-     signs  m 

division, 

ready  laid  down. 


ZERO     AND     INFINITY. 

§  306.  The  terms  zero  and  infinity  have  given    zero  and 

Infinity. 

rise  to  much  discussion,  and  been  regarded  as 
presenting  difficulties  not  easily  removed.  It  may 
not  be  easy  to  frame  a  form  of  language  that  shall 
convey  to  a  mind,  but  little  versed  in  mathe- 
matical science,  the  precise  ideas  which  these 
terms  are  designed  to  express ;  but  we  are  un- 
willing to  suppose  that  the  ideas  themselves  arc 
beyond  the  grasp  of  an  ordinary  intellect.  The 
terms  are  used  to  designate  the  two  limits  of 
Space  and  Number. 


Ideas  not 
abstruse. 


§  307.  Assuming  any  two  points  in  space,  and 
joining  them  by  a  straight  line,  the  distance  be- 
tween the  points  will  be  truly  indicated  by  the 
length  of  this  line,  and  this  length  may  be  ex- 
pressed numerically  by  the  number  of  times 
which  the  line  contains  a  known  unit.  If  now, 
the  points  are   made  to  approach  each  other,  the    ninstratiou, 

showing  the 

length  of   the  line  will   diminish   as   the   points    meaning  of 
come  nearer  and  nearer  together,  until  at  length,       ,^g^j,_ 
when  the  two  points  become  one,  the  length  ot 
the  line  will  disappear,  having  attained  its  limit. 


278  MATHEMATICAL     SCIENCE.  [bOOK  II 


-     which  is  called  zero.     If,  on   the  contrary,  the 

points  recede  from  each  other,  the  length  of  the 

Illustration,  line  joining  them  will  continually  increase  ;  but 

showing  the 

meaning  of   SO  loug  as  the  length  of  the  line  can  be  expressed 

the  term       .  c         i  •        c  •      • 

infiuity.  1^  terms  oi  a  known  unit  oi  measure,  it  is  not 
infinite.  But,  if  we  suppose  the  points  removed, 
so  that  any  known  unit  of  measure  would  occupy 
no  appreciable  portion  of  the  line,  then  the  length 
of  the  line  is  said  to  be  Infinite. 

§  308.  Assuming  one  as  the  unit  of  number, 

and  admitting  the  self-evident  truth  that  it  may 

be    increased    or  diminished,  we    shall  have  no 

zero^and  In-  difficulty    in    Understanding   the    import    of   the 

tinityapphed  terms  zero   and  infinity,  as   applied   to  number. 

to  numbers.  •'  ^  '■ 

For,  if  we  suppose  the  unit  one  to  be  continually 
diminished,  by  division   or  otherwise,   the  frac- 

luustratiou,  tioual  uuits  thus  arising  will  be  less  and  less, 
and  in  proportion  as  we  continue  the  divisions, 
they  will  continue  to  diminish.  Now,  the  limit 
or  boundary  to  which  these  very  small  fractions 
Zero:  approach,  is  called  Zero,  or  nothing.  So  long 
as  the  fractional  number  forms  an  appreciable 
part  of  one,  it  is  not  zero,  but  a  finite  fraction  ; 
and  the  term  zero  is  only  applicable  to  that 
which  forms  no  appreciable  part  of  the  standard. 

uiustraiion.  If,  ou  the  Other  hand,  we  suppose  a  numbei 
to  be  continually  increased,  the  relation  of  this 


CHAP,   IV."]  ALGEBRA,  279 

number  to  the  unit  will  be  constantly  changing. 
So  long  as  the  number  can  be  expressed  in 
terms  of  the  unit  one,  it  is  finite,  and  not  infi-  infinity; 
nite ;  but  when  the  unit  one  forms  no  appre- 
ciable part  of  the  number,  the  term  infinite  is 
used  to  express  that  state  of  value,  or  rather, 
that  limit  of  value. 

§  309.  The  terms  zero  and  infinity  are  there-   xheierms; 
fore  employed  to  designate  the  limits  to  which    employed, 
decreasing    and   increasing    quantities    may    be 
made  to    approach   nearer  than   any  assignable 
quantity ;  but  these  limits  cannot  be  compared,    Are  limits, 
in  respect  to  magnitude,  with  any  known  stand- 
ard, so  as  to  give  a  finite  ratio. 

§  310.  It  may,  perhaps,  appear  somewhat  par-  -vvhyiimits-i 
adoxical,  that  zero  and  infinity  should  be  defined 
as  "  the  limits  of  number  and  space"  when  they 
•are  in  themselves  not  measurable.     But  a  limit 
is  that  "  which  sets  bounds  to,  or  circumscribes  ;"  Definition  of 
and  as   all  finite  space  and  finite  number   (and      ^  '™' ' 
such  only  are  implied  by  the  terms  Space  and  of  Space  and 
Number),  are   contained  between  zero   and  in-       """  °' 
finity,  we  employ  these  terms  to  designate  the 
limits  of  Number  and  Space. 


280  MATHEMATICAL     SCIENCE.  [boOK  II. 


OF     THE     EQUATION. 

ne.iuo;ive        g  311.   We  have  seen  that  all  deductive  rea- 

reasoning.  ... 

soning  involves  certain  processes  of  comparison, 

and  that  the  syllogism  is  the  formula  to  which 

those  processes   may  be   reduced.*     It   has  also 

Comparison  ^ecn  Stated  that  if  two  Quantities  be  compared 

of  quantities.  ' 

together,  there  will  necessarily  result  the  condi- 
condition.    tioii  of  equality  or  inequality.      The  equation  ii 
an  analytical  formula  for  expressing  equality. 

Subject  of         g  3;[2.     The   subject   of  equations  is  divided 

equations: 

how  divided,  into   two  parts.      The    first,  consists    in  finding 
First  part:    the  cquatiou  ;  that  is,  in  the  process  of  express- 
ing the  relations  existing  between  the  quantities 
considered,  by  means  of  the  algebraic   symbols 
statement,    and  fomiula.      This  is  called  the   Statement  of 
Second  part :  the  proposition.     The    second   is  purely  deduc- 
tive, and  consists,  in  Algebra,  in  what  is  called 
Solution,     the  solution  of  the  equation,  or  finding  the  value 
of  the   unknown    quantity ;    and    in    the    other 
branches  of  analysis,  it  consists  in  the    discus- 
uiscussion  of  sioii  of  the  equatiou  ;  that  is,  in  the  drawing  out 
anequaion.  ^^^^  ^^^  equatiou  cvciy  thing  which  it  is  ca- 
pable of  expressing. 

»  Section  ICO. 


CHAP.    IV.]  ALGEBRA.  281 

§  313.  Making  the  statement,  or  finding  the  statement: 
equation,  is  merely  analyzing  the  problem,  and  what  it  is. 
expressing  its  elements  and  their  relations  in 
the  language  of  analysis.  It  is,  in  truth,  col- 
lating the  facts,  noting  their  bearing  and  con- 
nection, and  inferring  some  general  law  or  prin- 
ciple which  leads  to  the  formation  of  an  equation. 

The  condition  of  equality  between  two  quan-  Equniity  or 
tities  is  expressed  by  the  sign  of  equality,  which  ^^^^" ' 
is  placed  between  them.     The  quantity  on  the     How  ex- 

....  ^  pressed. 

left  of  the  sign  of  equality  is  called  the  first  mem-  .  ,       , 

iD  n.  J  1st  member, 

her,  and  that  on  the  right,  the  second  member  2d  member, 
of  the  equation.     The  first  member  corresponds 
to  the  subject  of  a  proposition  ;  the  sign  of  equal-     subject. 
ity  is  copula  and  part  of  the  predicate,  signify-    Predicate, 
ing,  IS  EQUAL  TO.     Hence,  an  equation  is  merely 
a  proposition    expressed   algebraically,  in  which  Pi-^'position. 
equality   is  predicated  of  one  quantity  as  com- 
pared with  another.     It  is   the  great  formula  of 
analysis. 

§  314.  We  have  seen  that  every  quantity  is    Abstract. 
either  abstract  or  concrete  :*    hence,  an   equa-    concrete. 
tion,  which  is  a  general  formula  for  expressing 
equality,  must  be  either  abstract  or  concrete. 

An    abstract    equation    expresses    merely    the 

*  Section  78. 


283 


MATHEMATICAL     SCIENCE.  [bOOK  11 


relation  of  equality  between  two  abstract  quan 
tities  :  thus, 

a  -i-b  =  X, 


Abatr.nct 
equation. 


Concrete 
equation. 


is  an  abstract  equation,  if  no  unit  of  value  be 
assigned  to  either  member ;  for,  until  that  be 
done  the  abstract  unit  one  is  understood,  and  the 
formula  merely  expi'esses  that  the  sum  of  a  and  h 
is  equal  to  x,  and  is  true,  equally,  of  all  quantities. 
But  if  we  assign  a  concrete  unit  of  value,  that 
is,  say  that  a  and  b  shall  each  denote  so  many 
pounds  weight,  or  so  many  feet  or  yards  of 
length,  X  will  be  of  the  same  denomination,  and 
the  equation  will  become  concrete  or  denominate. 


Five  opera-  §  315.  We  havc  secu  that  tiiere  are  six  oper- 
performed.  ations  which  may  be  performed  on  an  algebraic 
quantity.*  We  assume,  as  an  axiom,  that  if 
the  same  operation,  under  either  of  these  pro- 
cesses, be  performed  on  both  members  of  an 
equation,  the  equality  of  the  members  will  not  be 
changed.     Hence,  we  have  the  five  following 


Axioms. 


First. 


AXIOMS. 

1.  If  equal  quantities  be  added  to  both  mem- 
bers of  an  equation,  the  equality  of  the  members 
will  not  be  destroyed. 

*  Section  392. 


CHAP.  IV.  ] 


ALGEBRA. 


283 


2.  If  equal  quantities  be  subtracted  from  both     secomi 
members  of  an  equation,  the  equahty  will  not  be 
destroyed. 

3.  If  both  members  of  an  equation  be  multi-      Thw. 
plied  by  the  same  number,  the  equality  will  not 

be  destroyed. 

4.  If  both  members  of  an  equation  be  divided     Fourth, 
by  the   same  number,   the  equality   will  not  be 
destroyed. 

5.  If  both  members  of  an  equation  be  raised  to 
the  same  power,  the  equality  of  the  members  will 
not  be  changed. 

G.  If  the  same  root  of  both  members  of  an 
equation  be  extracted,  the  equality  of  the  mem- 
bers will  not  be  destroyed. 

Every  operation  performed  on  an  equation 
will  fall  under  one  or  other  of  these  axioms,  and 
they  afford  the  means  of  solving  all  equations 
which  admit  of  solution. 


Fifth. 


Sixth. 


Use  of 
axioms. 


§  316.  The  term  Equal,  in  Algebra,  implies  that      Equal, 
each  of  the  two  quantities  of  which  it  is  predi-  it^J^eaning 

^  -^       m  Algebra. 

cated,  contains  the  same  unit  an  equal  number  of 

times.  So  in  Geometry,  two  figures  are  equal  when  its  meaning 

,1  ,    •        n  -IP  1     in  Geome- 

they  contain  the  same  unit  of  measure  an  equal 
number  of  times.  If  in  addition  to  this  equality 
of  measure,  they  are  capable  of  superposition,  they 

.  Equal  in  all 

are  then  said  to  be  equal  in  all  their  parts.  parts. 


284  MATHEMATICAL     SCIENCE.  [bOOK  II 

§  317.  We  have  thus  pointed  out  some  of  tlie 

marked  characteristics  of  analysis.     In  Algebra, 

ciaswsof     the    elementary   branch,    the    quantities,    about 

qu.'intitie^ '"         i  •    i        i  •  •  i  •     •  j     j 

Algebra,  which  the  scieucc  IS  conversant,  are  aiviaea, 
as  has  been  already  remarked,  into  known  and 
unknown,  and  the  connections  between  them, 
expressed  by  the  equation,  afford  the  means  of 
tracing  out  further  relations,  and  of  finding  the 
values  of  the  unknown  quantities  in  terms  of  the 
known. 

In  the  other  branches  of  analysis,  the  quanti- 
How  divided  tJes    considered    are    divided    into    two   general 

in  the  other 

branches  of  classcs.  Constant  and  Variable  ;  the  former  pre- 

Analysis. 

servnig  fixed  values  throughout  the  same  pro- 
cess of  investigation,  while  the  latter  undergo 
changes  of  value  according  to  fixed  laws ;  and 
from  such  changes  we  deduce,  by  means  of  the 
equation,  common  principles,  and  general  prop- 
erties applicable  to  all  quantities. 


reasoning 

accounted 

for. 


Correspond-       §  318.   The  Correspondence  between  the  pro- 

ence  in  .  i  -i  •       i   ■         i 

methods  of  ccsscs  01  reasoumg,  as  exhibited  in  the  subject  of 
general  logic,  and  those  which  are  employed  in 
mathematical  science,  is  readily  accounted  for, 
when  we  reflect,  that  the  reasoning  process  is 
essentially  the  same  in  all  cases ;  and  that  any 
change  in  the  language  employed,  or  in  the  sub- 
ject to  which  the  reasoning  is  applied,  does  noi 


CHAP.   IV.]  ALGEBRA.  2<Sn 

al  all  change  the  natui'e  of  the  process,  or  mate- 
rially vary  its  form. 

§  319.  We  shall  not  pursue  the  subject  of 
algebra  any  further ;  for,  it  would  be  foreign 
to  the  purposes  of  the  present  work  to  attempt    objects  of 

,  .  ,  ,     ^  ,     the  present 

more  than  to  pomt  out  the  general  leatures  and  ^0^^: 
characteristics  of  the  different  branches  of  math- 
ematical science,  to  present  the  subjects  about 
which  the  science  is  conversant,  to  explain  the 
peculiarities  of  the  language,  the  nature  of  the 
reasoning  processes  employed,  and  of  the  con- 
necting links  of  that  golden  chain  which  binds  exteiided. 
together  all  the  parts,  forming  an  harmonious 
whole. 


SUGGESTIONS  FOR  THOSE  WHO  TEACH  ALGEBRA. 


1.  Be  careful  to  explain  that  the  letters  em-    Letters 

but  ni( 
sjinbola. 


are 


ployed,  are  the  mere  symbols  of  quantity.     That      "  ™^'^'' 


of,  and  in  themselves,  they  have  no  meaning  or 
signification  whatever,  but  are  used  merely  as 
the  signs  or  representatives  of  such  quantities 
as  they  may  be  employed  to  denote. 

2.  Be  careful  to  explain  that  the  signs  which    signs  indi- 

,  ,  ,  ^       r  \  cateoper» 

are   used  are  employed  merely  lor  the  purpose       uoua. 
of  indicating  the    six    operations  which   may  be 
performed  on  quantity ;  and  that  they  indicate 


MATHEMATICAL     SCIENCE, 


[book  II. 


operations  merely,  without  at  all  affecting  the 
nature  of  the  quantities  before  which  they  are 
placed. 

3.  Explain  that  the  letters  and  signs  are   the 
elements  of  the  algebraic  language,  and  that  the 


Letters  and 

signs 
elements  of 
language. 


language  itself  arises  from   the   combination  of 
these  elements. 

4.  Explain  that  the  finding  of  an  algebraic 
formula  is  but  the  translation  of  certain  ideas, 
first  expressed  in  our  common  language,  into 
the  language  of  Algebra ;  and  that  the  interpre- 
tation of  an  algebraic  formula  is  merely  transt 
lating  its  various  significations  into  common 
language. 

5.  Let  the  language  of  Algebra  be  carefully 
studied,  so  that  its  construction  and  significa- 
tions may  be  clearly  apprehended. 

6.  Let  the  difference  between  a  coefficient 
and  an  exponent  be  carefully  noted,  and  the 
office  of  each  often  explained ;  and  illustrate  fre 
quently  the  signification  of  the  language  by  at- 
tributing numerical  values  to  letters  in  various 
algebraic  expressions. 

7.  Point  out  often  the  characteristics  of  sim- 
ilar and  dissimilar  quantities,  and  explain  which 
may  be  incorporated  and  which  cannot. 

Minus  sign.        g.  Explain  the  power  of  the  minus  sign,  as 
shown  in   the  four  ground  rules,  but  very  par- 


Algebraic 
formula : 


Its  interpret- 
ation. 


Language, 


Coefficient, 
Exponent. 


Similar 
quantities. 


CHAP.   IV.]  ALGEBRA.  287 

ticularly  as  it  is  illustrated  in  subtraction,  multi- 
,  plication,  and  division. 

9.  Point  out  and  illustrate  the  correspondence 
between    the    four   ground   rules    of  Arithnaetic    Arithmetic 

and   Algebra 

and  Algebra;  and  impress   the  fact,   that   their    compared, 
differences,  wherever  they  appear,  arise  merely 
from  differences  in  notation  and  language  :  the 
principles   which    govern    the    operations    being 
the  same  in  both. 

10.  Explain  with  great  minuteness    and  par-    Equation, 
ticularity  all  the  characteristic  properties  of  the    I's  proper 

lies. 

equation  ;  the  manner  of  forming  it ;  the  differ- 
ent kinds  of  quantity  which  enter  into  its  com- 
position ;  its  examination  or  discussion  ;  and 
the  different  methods  of  elimination. 

11.  In  the  equation  of  the  second  degree,  be  Equations 

^  ,  1         11  1         r-  r  1-1  *li®  second 

careful   to  dwell  on   the  four  forms  which  em-      degree. 
brace  all  the  cases,  and  illustrate  by  many  ex- 
amples that  every  equation  of  the  second  de- 
gree may  be  reduced  to  one  or  other  of  them,     its  forms. 
Explain   very  particularly  the    meaning  of  the 
term  root ;  and  then  show,  why  every  equation     ^^  roots. 
of  the  first  degree  has  one,  and  every  equation 
of  the  second  degree,  two.     Dwell  on  the  prop- 
erties of  these  roots  in  the  equation  of  the  sec- 
ond  degree.     Show  why   their    sum,  in   all   the    Their  sum, 
forms,  is  equal  to  the  coefficient  of  the  second 
term,  taken  with  a  contrary  sign ;  and  why  their 


288  MATHEMATICAL     SCIENCE.  [bOOK  II. 

Their  prod-  product  is    cqual   to   the   absolute   term  with   a 
"*^'*        contrary  sign.     Explain  when  and  why  the  roots 
are  imaginary. 
General  12.    In  fine,   remember  that  every  operation 

Principles :  .        .    ,         ^         . 

and  rule  is  based  on  a  principle  of  science,  and 

that  an  intelligible  reason  may  be  given  for  it. 

Find   that  reason,  and  impress   it   on   the  mind 

Should  be    of  your  pupil  in  plain  and  simple  language,  and 

exp  ainc  .    ^^  familiar  and   appropriate   illustrations.     You 

will    thus  impress  right   habits  of  investigation 

and  study,  and  he  will  grow  in  knowledge.     The 

broad  field   of    analytical    investigation   will    be 

opened   to  his    intellectual   vision,    and    he    will 

have  made  the  first  steps  in  that  sublime  science 

They  lead  to  which  discovcTS  the  laws  of  nature  in  their  most 

general  laws. 

secret  hiding-places,  and  follows  them,  as  they 
reach  out,  in  omnipotent  power,  to  control  the 
motions  of  matter  through  the  entire  regions  of 
occupied  space. 


CHAP,   v.]  DIFPEKENTIAL    CALCULUS.  289 


CHAPTER   V. 


DIPFERENTIAX.    CALCULUS. 


§  330.   The   entire   science  of   mathematics  is    Science  of 

Mathema- 
conversant  about   the  properties,   relations,  and       tics. 

measurement  of  quantity.     Quantity  has  already 

been  defined.     It  embraces  everything  which  can 

be  increased,  diminished,  and  measured. 

In   the   elementary  branches  of  mathematics,  Discontina- 

0U9  quan- 

quantity  is  regarded  as  made  up  of  parts.  If  the  tity. 
parts  are  equal,  each  is  called  a  unit,  and  the 
measure  of  a  quantity  is  the  number  of  times 
which  it  contains  its  unit.  Such  quantities  are 
called  discontinuous ;  because,  in  passing  from 
one  state  of  value  to  another,  we  go  by  the  steps 
of  the  unit,  and  hence,  pass  over  all  values  lying 
between  adjacent  units. 

Thus,  if  Ave  increase  a  line  from  one  foot  to  Example  in 

disconlina- 

forty  feet,  by  the  continued  addition  of  one  foot,    ous  quan- 
tity, 
we  touch  the  line,  in  our  computation,  only  at  its 

two  extremities,  and  at  thirty-nine  intermediate 

points,  of  which  any  two  adjacent  points  are  one 

foot  apart.     In  the  scale  of  ascending  numbers', 

1,  2j  3,  4,  5,  6,  etc.,  Ave  pass  over  all  quantities  less 

19 


290  MATHEMATICAL     SCIEN"CE.  [BOOK   II. 

than  that  Avhich  is  denoted  by  the  unit,  one. 
Discontinuous  quantities  are  generally  expressed 
by  numbers,  or  by  letters,  which  stand  for 
numbers. 

Continuous  §  321.  In  the  higher  branches  of  mathematics, 
the  laws  which  regulate  and  determine  the 
changes  of  quantity,  from  one  state  of  value  to 
another,  are  quite  different.  Suppose,  for  ex- 
ample, that  instead  of  considering  a  right  line  to 
^^  be  made  up  of  forty  feet,  or  of  480  inches,  or  of  960 
half  inches,  or  of  1920  quarter  inches,  or  of  any 
number  of  equal  parts  of  the  inch,  we  regard  it 
as  a  quantity  having  its  origin  at  0,  and  increas- 
ing according  to  such  a  law,  as  to  pass  through 
or  assume,  in  succession,  all  values  between  0  and 

Discontinu- 
ous,      forty  feet.     This  supposition  gives  us  the  same 

distance  as  before,  but  a  very  different  law  of  for- 
mation. A  quantity  so  formed  or  generated,  is 
called  a  continuous  quantity.     Hence, 

A  DiscoxTiKUOUS  QUANTITY  is  One  which  is 

Dincontinu-  made  up  of  parts,  and  in  which  the  changes,  in 
passing  from  one  state  of  value  to  another,  can  be 
expressed  in  numbers,  either  exactly,  or  approxi- 
mately ;  and 

A    CONTINUOUS    QUANTITY   is    one  which  in 

Continuous,  changing  from  one  state  of  value  to  another, 
according    to   a  fixed    law,    passes    through    oi 


CHAP,   v.]  DIFFERENTIAL    CALCULUS. 


291 


assumes,  in  succession,  all  the  intermediate 
values. 

Thus,  the  time  whicli  elapses  between  12  and  Space  cob- 

tinuoaa 

1  o'clock,  or  between  any  two  given  periods,  is 
continuous.  All  space  is  continuous,  and  every 
quantity  may  be  regarded  as  continuous,  which 
can  be  subjected  to  the  required  laAV  of  change. 


LIMITS. 

§  322.    The  limit  of  a  variable  quantity  is  a     Limits, 
quantity  towards  which  it  may  be  made  to  ap- 
proach  nearer    than    any    given    quantity,   and 
which  it  reaches,  under  a  particular  supposition. 


limits  of  discontinuous  quantity. 

§  323.    The  limits  of  a  discontinuous  quantity     General 
are  merely  numerical  boundaries,  beyond  which 
the  quantity  cannot  pass. 

For  positive  quantities,  the  minimum  limit  is     Limits. 
0,  and  the  maximum  limit,  infinity.     For  nega- 
tive quantities,  they  are  0,  and  minus  infinity; 
and  generally,  using  the  algebraic  language,  the 
limits  of  all  quantities  are, 

Minimum  limit,  —  infinity;    maximum  limit, 
+  infinity. 

We  can   illustrate  these  limits,  and  also  what  Examples. 


293 


MATHEMATICAL     SCIEKCE.  [BOOK   II. 


we  mean  by  the  terms,  0  and  infinity,  plus  or 
minus,  by  reference  to  the  trigonometrical  func- 
tions. Thus,  when  the  arc  is  0,  the  sine  is  0. 
When  the  arc  increases  to  90°,  the  sine  attains 
its  maximum  value,  the  radius,  E.  Passing  into 
Einstra-     the  second  quadrant,  the  sine  diminishes  as  the 

lions. 

arc  increases,  and  when  the  arc  reaches  180°,  the 
sine  becomes  0.  From  that  point,  to  270°,  the 
sine  increases  numerically,  but  decreases  algebraic- 
ally,  and  at  270°,  its  minimum  value  is  —  R, 
From  270°  to  360°,  the  sine  decreases  numeric- 
Limits,  ally,  but  increases  algebraically.  Hence,  the  nu- 
merical limits  of  the  sine,  are  0  and  B;  and  its 
algebraic  limits,  —  R  and  +  R. 

Let  us  now  consider  the  tangent.  For  the  arc 
Tangent.  0,  the  tangent  is  0.  If  the  arc  be  increased  from 
0  towards  90°,  the  lengtli  of  the  tangent  will 
increase  and  as  the  arc  approaches  90°,  the  pro- 
longed radius  or  secant  becomes  more  nearly  par- 
allel with  the  tangent;  and  finally,  at  90°  it 
becomes  absolutely  parallel  to  it,  and  the  length 
of  the  tangent  becomes  greater  than  any  assign- 
alle  line.  Then  we  say.  that  the  tangent  of  90° 
is  infinite  ;  and  we  designate  that  quantity  by  oo  . 
After  90°,  the  tangent  becomes  minus,  and  con- 
tinues so  to  the  end  of  the  second  quadrant, 
where  it  becomes  —  0;  and  at  270°  it  becomes 
equal  to,  +  oo .     The  secant  of  90°  is  also  equal  to 


CHAP,   v.]  DirFEREKTIAL    CALCULUS.  293 

+  00 ;  and  of  270°,  to  —  oo .     These  illustrations    When  it 
indicate  the  significations  of  the  teniis,  zero  and 
infinity.     Tliey  denote  t\\e.  limits  towards  which   reaches  itB 
variable   quantities    may   be   made   to   approach 
nearer  than  any  given  quantity,  and  which  limits     Limits, 
are  reached  under  particular  suppositions. 

§  324.   The  term,  given,  or  assignable  quantity,    Quantity, 
denotes  any  quantity  of  a  limited  and  fixed-  value. 

The  term,  infinitely  great,  or  infinity,  denotes     infinite. 
a  quantity  greater  than  any  assignable  quantity 
of  the  same  kind. 

The  term,  infinitely  small,  or  infinitesimal,  de-    infinitesl- 

nal. 
notes  a  quantity  less  than  any  assignable  quan- 
tity of  the  same  kind. 


COifTIKUOUS   QUANTITIES. 

§  325.  A  continuous  quantity  has  already  been  Continuous, 
defined  (Art.  325).     By  its  definition  it  has  two 
attributes : 

1st.  That  it  shall  change  its  value  according  to    Quantity. 
a  fixed  law  ;  and 

2d.  That  in  changing  its  value,  between  any  Attributes, 
two  limits,  it  shall  pass  through  all  the  interme- 
diate values. 

§  32G.   CoxsECUTiVE  Values. — Two  values  of  consecutive. 


294 


MATHEMATICAL     SCIENCE.  [BOOK    II. 


a  coDtinuous  quantity  are  consecutive  when,  if 
the  greater  be  diminished,  or  the  less  increased, 
according  to  the  Imv  of  cliange,  the  two  values 
will  become  equal. 

Let  A  be  the  origin  of  a  system  of  rectangu- 
lar co-ordinate  axes,  and  C  a  given  point  on  the 
axis  of  X. 

If  we  suppose  a  point  to  move  from  A, 
in  the  plane  of  the  axes,  and  with  the  farther 
condition,  that  it  shall  continue  at  the  same 
distance  from  the  point  C,  it  will  generate  the 
circumference  of  a  circle,  APBDEA,  beginning 
and  terminating  at  the  point  A.  The  moving 
point  is  called  the  generatrix. 


Quantities 
generated. 


dirde. 


The  circumference  of  this  circle  may  also  be 
generated  in  another  way,  thus  : 

Denote  the  straight  line  AD  by  'ZR,  and  sup- 


line  and 


CHAP,   v.]  DIFFEEEKTIAL    CALCULITS.  295 

pose  a  point  to  move  uniformly  from  A  to  D. 

Denote  the   distance   from  A    to   any  point   of 

the   line   AD,   hjx:   then,    the    other    segment     Tangent 

will  be  denoted  by  2]i  —  x.     ISow,  at  evei^y point 

of  AD,    su]opose   a   perpendicular   to   be   drawn 

to  AD.      Denote  each  perpendicular  by  y,  and 

suppose  y  always  to   have   such  a  value  as  to 

satisfy  the  equation  Equation. 

if  ^  2Rx-x\ 

Under  these  hypotheses,  it  is  plain  that  the  ex-  supposition 

tremities  of  the  ordinate  y  will  be  found  in  the 

circumference   of    the   circle,  which    will    be    a 

continuous  quantity.      The    ordinate   y  will  be 

contained,   in    the    first    quadrant,   between   the      on  the 

numerical  limits  of  ?/  =  0  and  y  =  +R',  in  the 

second,  between    the    numerical    limits   of  y  = 

+  B,  and  y  =  0 ;  in  the  third,  between  y  =  —  0 

and  y  =  —  E;  and  in  the  fourth,  between  y  =z    Equation. 

—  R  and  y  —  —  0. 

The  circumference  ABDEA,  may  be  regarded 

under  two  points  of  view: 

First.  As  a  discontinuous   quantity,  expressed  When  con- 
tinuous. 

in  numbers :  viz.  hy  AD  X  3.1416  ;  or  it  may  be 
expressed  in   degrees,   minutes,   or   seconds,  viz, 
360°,  or  21600',  or  1296000".     In  the  first  case,    istcase. 
the  step,  or  change,  in  passing  from  one  value  to 
the  next,  will  be  the  unit  of  the  diameter  AD. 


296  MATHEMATICAL     SCIEIfCE.  [iSOOK   II. 

2d  Case.  In  the  second,  it  Av  ill  be  one  degree,  one  minute, 
or  one  second.  In  neither  case,  will  the  parts  of 
the  circumference  less  than  the  unit  be  reached 
by  the  computation.     Or, 

Secondly.  Secondly:  AVe  may  regard  the  cii'cumference 
as  a  continuous  quantity,  beginning  and  termi- 
nating at  A.     Under  this  supposition,  the  gene- 

whendis-  ratrix  will  occupy,  in  succession,  every  point  of 

continuous.    ,,  .  „  i-n-  -l' 

the   circumierence,   and   will,   m   every  position, 
satisfy  the  equation 

y-  —  2Ex  —  x^. 


Idfinitesi- 
mal. 


Hence,  if  we  measure  a  quantity  by  a  finite 
unit,  that  quantity  is  discontinuous;  but  if  we 
measure  it  by  an  wfinitesimal  unit,  the  quantity 
becomes  continuous. 


TAKGENT   LINE   AKD   LIMIT. 

§  327.   Take  any  point  of  the  circumference  of 

Tangent    this  circle,  as  P,  whose  co-ordinates  are  x'  and  y', 

^™' '      and  a  second  point  JI,  Avhose  co-ordinates  are  x" 

Secant  line,  and  y",  and  through  these  points  draw  the  secant 

line,  HP  G. 

Then,  HJ  =  y"  -  y',  and  PJ  =  x"  -  x' ;  and 

H£_^  f-y'  ^  tang,  of  the  angle  HP  J,  or 
PJ      x"-x'  *=  *" 

HQC. 


CHAP,   v.]  DIFFERENTIAL    CALCULUS. 


297 


becomes  a 


Let  lis  now  suppose  a  tangent  line  TP  to  be  when  it 
drawn  to  the  circle,  toucliing  it  at  P.  If  we 
suppose  the  point  H  to  approach  the  point  P,  it 
is  plain  that  the  value  of  y"  will  approach  to 
the  value  of  y',  and  the  value  of  x"  to  that  of  x' ; 
and  when  the  point  H  becomes  consecutive  with 
the  point  P,  y"  and  y'  will  become  consecutive, 
and  so  also  Avill  x"  and  x'.  When  the  point  H 
becomes  consecutive  with  the  point  P,  the  secant 
line,  HG,  becomes  the  tangent  line  PT.  For, 
since  the  arc  is  a  continuous  quantity,  no  point 
of  it  can  lie  between  two  of  its  consecutive 
values ;  and  hence,  at  P,  no  point  of  the  curve 
can  lie  above  the  line  TP ;  therefore,  by  the  de- 
finitions of  Geometry,  TP  is  a  tangent  line  to 
the  circle  at  the  point  P. 


tangent. 


But  the  definition  of  a  tangent  line  to  a  circle,  Tangent  of 


298  MATHEMATICAL     SCIEJfCE.  [BOOK    II. 

Elementary  ill  elementary  Geometry,  viz.  that  it  touches  the 

Geometry. 

circumference  in  one  point,  is  incomplete.  It 
is  provisional  only.  For,  as  we  now  see,  the 
tangent  line  touches  the  circle  in  two  consecu- 
Positionof  H'v^  J90i«/s,  Avhich,  in  discontinuous  quantity, 
are  regarded  as  one,  because  the  distance  between 
them,  expressed  numerically,  is  zero. 

If  we  prolong  JH  till  it  meets  the  tangent  line 
at  0,  we  see  that, 

Secant.      —^ —  =  tangent  of  OP  J  =  tangent  of  OTO; 

and  that, 

— ^^ — ^  =  tangent  of  HPJ  =  tangent  of  HGC. 

When  the  point  H  approaches  the  point  P 
Wheuthe  nearer  than  any  given  distance,  the  angle  HGG 
will  approach  the  angle  PTC  nearer  than  any 
given  angle,  and  when  H  becomes  consecutive 
Secant  y^r[^\^  p^  ^j^g  angle  HGC  will  become  equal  to 
tlie  angle  PTC,  which  is  therefore  its  limit. 
Under  this  hypothesis,  the  point  H  falls  on  the 
tangent    line,    and  JH  becomes    equal    to   JO. 

becomes 

Under  the  same  hypothesis,  y  and  y  become 
consecutive,  and  also  x'  and  x' ;  hence,  y"  —  y' 
becomes  less  than  any  given  quantity;  and  so, 
a  tangent  ^^^°'  does  ;?;"  —  x'.  This  diflerence  between  con- 
secutive values  is  expressed  by  simply  writing 
the  letter  d  before  the  variable.     Thus,  the  dif- 


CHAP.   V.J  DIFFERENTIAL    CALCULUS. 


299 


ference  of  the  consecutive  Tallies  of  y  is  de- 
denoted  by  cly  ]  and  is  read  differential  of  y ; 
and  the  difierence  between  the  consecutive 
values  of  x  is  denoted  by  dx,  and  is  read  differ- 
ential of  X.     Hence,  "\ve  have 

-^  =  tangent  PTC:  viz. 
dx  ^  ' 

the  tangent  of  the  angle  which  the  tangent  at 
the  point  P  makes  the  axis  of  X. 

By  the  definition  of  a  limit,  dy  becomes  the 
limit  of  y"—  y',  and  dx  the  limit  of  x"—  x',  under 


Value 


oftho 


Angle. 


Limit. 


the  supposition  that  i/"  and  ?/',  and  .'c"  and  a;' be-  when  it  ia 

^  '  '^  •'  _      _         reached. 

come,  respectively,  consecutive.  The  term  limit, 
therefore,  used  to  designate  the  ultimate  differ- 
ence between  two  values  of  a  variable,  denotes 
the   actnal   difference   between   its  two  consecu- 

Inflnitely 

tive   values;   this   difference   is   infinitely   small,      email. 


300  MATHEMATICAL    SCIENCE.  [BOOK   II. 

and  consecutive  with  zero.  For,  if  after  y"  has 
become  consecutiye  with  y',  it  be  again  dimin- 
ished, according  to  tlie  law  of  change  expressed 
by  the  equation 

Equation.  y^  =:  '^Rx  —  x\ 

wheuthe  it  will,  fi'oni  the  definition  of  consecutive  values, 
become  equal  to  y',  and  then  x"  will  become  equal 
x',  and  Ave  shall  have 

tangent  y"  ^  y'  -  Q  and  x"  -  x'  =  {). 

Under  this  hypothesis  the  line  PT  has,  at  P, 
becomes  a  but  a  Single  ])oinf,  common  with  the  circumfer- 
ence of  the  circle  ;  it  then  ceases  to  be  a  tangent, 
and  becomes   any   secant   line   passing   through 
Secant.     ^^^^^  point  and  intersecting  the  circumference  in 
a  second  point. 


Generally 
true. 


§  328.  Wliat  we  have  here  shown  in  regard  to 
the  circumference  of  the  circle,  and  its  tangent, 
is  equally  true  of  any  other  curve  and  its  tan- 
gent, as  may  be  shown  by  a  very  slight  modifica- 
tion of  the  process. 
A  straight       The  fact,   that   a   straight   line   tangent   to   a 

line;  when  i  i  i.-  •    j.  -j.! 

a  tangent  curve,  has  two  cousecutive  points  common  Avith 
it,  appears  in  all  the  elementary  problems  of 
tangents.  The  conditions  are,  an  equality  be- 
tween the  co-ordinates  of  the  point  of  contact 


CHAP,    v.]  DIFFEEEIS'"TIAL     CALCULUS.  301 

and  the  first  differential  co-efficients,  at  tlie  same 
point,  of  the  straight  line  and  curve.  These  con- 
ditions fix  the  consecutive  points  common  to  the 
straight  line  and  curve. 

Analysis,  therefore,  by  its  searching  and   mi-    Anaiyeia; 
croscopic  powers — by  looking  into  the   changes 
which  take  place  in  quantity,  as  it  passes  from 
one  state  of  value   to  another,  develops  proper-    its  power, 
ties  and  laws  which  lie  beyond  the  reach  of  the 
numerical  language.     Thus,  the  distance  between 
two  consecutive  points,  on  the  circumference  of 
a  circle,  cannot   be   expressed   by  numbers ;  for, 
however  small  the  number  might  be,  cliosen  to    Example, 
express  such  a  distance,  it  could  be  diminished, 
and  hence,  there  would  be  intermediate  points. 

The  introduction,  therefore,  of  continuous  quan-  continnous 

quantity; 

tity,  into  the  science  of  matliematics,  brought 
with  it  new  ideas  and  the  necessity  of  a  new 
language.  Quantity,  made  up  of  parts,  and  ex- 
pressed by  numbers,  is  a  veiy  difi'erent  thing 
from  the  continuous  quantity  treated  of  in  the 

Differential  and  Integral  Calculus.    Here,  the  law    What  fol- 
lowed its 
of  continuity,  in  the  change  from  one  state  of     introduc- 

tiou. 

value  to  another,  is  tlie  governing  principle,  and 

carries  with  it  many  consequences. 

Time  and  space  are  the  continuous  quantities    Time  and 

Space, 
with  which  Ave  are  most  conversant.     If  avc  take 

a  moment  in  time,  and  look  back  to  the  past,  or 


302 


MATHEMATICAL    SCIEN-GE,  [BOOK   II. 


forward  to  the  future,  there  is  no  interruption. 
The  law  of  continuity  is  unbroken,  and  the  iu- 
Exampies  finite  opens  to  our  contemplation.  If  we  take  a 
point  in  space,  and  through  it  conceive  a  straight 
line  to  be  drawn,  the  law  of  continuity  is  also 
there,  and  the  imagination  runs  along  it,  to  the 

of  the 

infinite,  in  either  direction.  The  attraction  of 
gravitation  is  a  continuous  force ;  and  all  the 
motions  to  which  it  gives  rise,  follow  the  law  of 
law  of  continuity.  All  growth  and  development,  in  the 
vegetable  and  animal  kingdoms,  so  far  as  we 
know,  conform  to  this  law.  This,  therefore,  is 
Continuity,  the  great  and  important  law  of  quantity,  and  the 
Higher  Calculus  is  conversant  mainly  about  its 
development  and  consequences. 


Coiipe- 
quences. 


First. 


Second. 


COK"SEQUE]SrCES   OF  THE    LAW   OF    COISTTIXUITY. 

1.  The  most  striking  consequence  of  the  law 
of  continuity,  is  the  fact,  that  whatever  be  the 
quantity  subjected  to  this  law,  or  whatever  be 
the  law  of  change,  the  difierence  between  any  two 
of  the  consecutive  values  is  an  infinitesimal, 
and  hence  cannot  be  expressed  by  numbers. 

2.  Since  a  continuous  quantity  may  be  of  any 
value,  and  be  subjected  to  any  law  of  change,  the 
infinitesimal  which  expresses  the  difference  be- 
tween any  two  of  its  consecutive  values,  is  a  vari- 


CHAP,   v.]  DIFFERENTIAL    CALCULUS. 


303 


aUe   quantity;   and   hence,  may   liave  any  yalue 
between  zero  and  its  maximum  limit. 

3.  Tlie   law  of  continuity  in  quantity,  there- 
fore, introduces  into  the  science  of  mathematics 

a  class  of  variables  called  infi)iitesimals,  or  differ-  Third. 
entials.  Every  variable  quantity  has,  at  every 
state  of  its  value,  an  infinitesimal  correspond- 
ing to  it.  This  infinitesimal  is  the  connecting 
link,  in  the  law  of  continuity,  and  will  vary 
Avith  the  value  of  the  quantity  and  the  law  of 
change, 

4.  In  the  Infinitesimal   Calculus,  the   proper- 
ties,  relations,   and    measurement   of    quantities 

are  developed  by  considering  the  laws  of  change     Fonrth. 
to  which   they  are   subjected.     The  elements  of 
the  language  employed,  are  symbols  of  those  in- 
finitesimals. 


NEWTO]S['S    METHOD    OP    TREATI^s^G    COK- 
TIKUOUS   QUANTITY.* 

Lemma  I. 

§  329.    Quantities,  and  the  ratios  of  quantities,     Rations 
lohich  in  any  finite  time  converge  continually  to 
equality,  and  iefore  the  end  of  that  time  approach        of 
7iearer,  the  one  to  the  other,  than,   hy  any  given 
difference,  become  ultimately  equal.  limits. 


*  Principiat  Book  I.,  Section  I. 


ao4 


MATHEMATICAL     SCIENCE. 


[book   II. 


If  you  deny  it,  suppose  tliem  to  be  ultimately 
unequal,  and  let  D  be  their  ultimate  difference. 
Therefore,  they  cannot  approach  nearer  to  equal- 
ity than  by  that  given  difference  D;  which  is 
against  the  supposition. 


Statement 


of  the 


a    I 


K 


m 


n 


Lemma  II. 

§  330.  If  in.  any  figure,  AacE,  terininated  hy 
rigid  lines  Aa,  AE,  and  the  curve  acE,  tliere  he 
inscribed  any  number  of  parallelograms  Ah,  Be,  Cd, 
etc.,  comprehended  under  the  equal  hases,  AB,  BC, 
CD,  etc.,  and  the  sides  Bh,  Cc,  Dd,  etc,  parallel 
to  one  side  Aa  of  the  fig- 
ure ;  and  the  parallelo- 
grams aKbl,  hLcm,  cMcln, 
d,DEo  are  completed;  then, 
if  the  breadth  of  these  par- 
generai  allelogravis  be  supposed  to 
he  diminished,  and  their 
number  to  be  augmented 

in  finitum ;  /  say,  that  the  ultimate  ratios 
Proposition,  which  the  inscribed  figicre  AKbLcMdD,  the  cir- 
cumscribed figure  AalbmcndoE  and  the  curvili- 
near figure  AabcdE  will  have  to  one  another,  are 
ratios  of  equality. 

For,  the  difference  of  the   inscribed  and   cir- 

Demonstra- 

tion       cumscribed  figures  is  the  sum  of  the  parallelo- 


d 


ABODE 


CHAP,   v.]  DIFFEEENTIAL    CALCULUS. 


305 


general 


grams,  Kl,  Lm,  Mn,  Do,  that  is,  (from  the  equal-  of  the 
ity  of  their  bases),  the  rectangle  under  one  of 
their  bases  Kb  and  the  sum  of  their  altitudes 
Aa;  that  is,  the  rectangle  ^5^a.  But  this  rect- 
angle, because  its  breadth  AB  is  supposed  dim- 
inished in  finitum,  becomes  less  than  any  given 
space.  And  therefore,  (by  Lemma  I.)  the  figures 
inscribed  and  circumscribed,  become  ultimately  Propoation. 
equal  one  to  the  other ;  and  much  more  will  the 
intermediate  curvilinear  figure  be  ultimately 
equal  to  either. 


Lemma  IIL 

§  331.    The  same  ultimate  ratios  are  also  ra-  statement 
tios  of  equality,  when  the  breadths  AB,  BC,  DC, 
etc.,  of  the  parallelograms   are   unequal  and  are 
all  diminished  in  finitum. 

For,  suppose  AF  to  be  the  greatest  breadth,  Demonstra- 
and  complete  the  paral- 
lelogram FAaf  This 
parallelogram  will  be 
greater  than  the  differ- 
ence of  the  inscribed  and 
circumscribed  figures  ; 
but  because  its  breadth 
^4i^is  diminished  infini- 

tum,  it  will   become  less   than  any  giyen  rect- 
angle. 

20 


tion  of  the 


BF~C 


306 


MATHEMATICAL    SCIENCE.  [BOOK   11. 


general         CoR.  1.   Hence,  the  ultimate  sum  of  these  evan- 
escent parallelograms  will,  iu  all  parts,  coincide 
Proposition.  Tvith  the  Curvilinear  figure. 

Cor.  2.  Much  more  will  the  rectilinear  figure 
Ultimately  Comprehended  under  the  chords  of  the  evanes- 
cent arcs,  ab,  he,  cd,  etc.,  ultimately  coincide  with 
the  curvilinear  figure. 

Cor.  3.  And  also,  the  circumscribed  rectilinear 
figure  comprehended  under  the  tangents  of  the 
same  arcs. 

Cor.  4.  And  therefore,  these  ultimate  figures 
(as  to  their  perimeters,  acE)  are  not  rectlinear, 
but  curvilinear  limits  of  rectilinear  figures. 


equal. 


Also, 
figures. 


Limit  of 
areas. 


Lemma  IV. 
statement       §  ^33.  If  ill  two  figures,  AacE,  PprT,  you  m- 
PropoBition  ^^'"^^^  (^-^  before)  two  ranhs  of  2Mrallelograms,  an 


Figure. 


eqiial  number  in  each  rank,  and,  where  their 
breadths  are  diminished,  in  finitum,  the  ultimate 
ratios  of  the  parallelograms  in  one  figure  to  those 
in  the  other,  each   to  each   respectively,  are  tlie 


CHAP,   v.]  DIFFERENTIAL    CALCULUS.  307 


same  ;  I  say,  that  those  two  figures,  AacE,  PprT, 

are  to  one  another  in  that  same  ratio. 

For,  as  the  parallelograms  in  the  one  fignre  are  Demonstra- 
tion, 
severally  to  the   parallelograms  in  the  other,  so 

(by  composition)  is  the  sum  of  all  in  the  one  to 
the  sum  of  all  in  the  other ;  and  so  is  the  one 
figure  to  the  other ;  because  (by  Lemma  III.), 
the  former  figure  to  the  former  sum,  and  the 
latter  figure  to  the  latter  sum,  are  both  in  the 
ratio  of  equality. 

Cor,  Hence,  if  two  quantities  of  any  kind  are  Theabove 
anyhow  divided  into  an  equal  number  of  parts, 
and  those  parts,  when  their  number  is  aug- 
mented, and  their  magnitude  diminished  in  fini- 
tum,  have  a  given  ratio  one  to  the  other,  the  first  P'«p««5«<"»» 
to  the  first,  the  second  to  the  second,  and  so  on 
in  order,  the  whole  quantities  will  be  one  to  the 

other  in  that  same  given  ratio.      For  if  in  the 

trae  for  all 

figures   of   tliis    lemma,   the    parallelograms   are 

taken  one  to  the  other  in  the  ratio  of  the  parts, 

the  sum  of  the  parts  will  always  be  as  the  sum  of 

tli8  parallelograms;  and  therefore,  supposing  the    kinds  of 

number  of  the  parallelograms   and  parts   to  be 

augmented,  and  their  magnitudes  diminished  in 

finitum,  those  sums  will  be  in  the  ultimate  ratio 

of  the  parallelogram  in  the  one  figure  to  the  cor-    ^"a^t^'y- 

responding  parallelogram  in  the  other;  that  is 

(by  the   supposition),  in  the   ultimate  ratio  of 


308 


MATHEMATICAL    SCIEXCE.  [BOOK   IL 


any  one  part  of  the  one  quantity  to  the  corre- 
sponding part  of  the  other. 


Lemma  V. 

§  333.   In  similm-  figures  all  sorts  of  Jiomolo- 
statement,  gous  Sides,  whether  curvilinear  or  rectilinear,  are 
proportional ;  and  their  areas  are  in  the  duplicate 
ratio  of  their  homologous  sides. 


Lemma  VL 

§  334.   If  the  arc  ACB,  given  in  position,  is 
statement  Subtended  hy  the  chord  AB,  and  in  any  point  A 

in  the  middle  of  the 

continued  curvature, 

is  touched  by  a  rigid 

line   AD,   produced 

both  ways;   then,  if 

the  points  A  and  B 

afproach  one  another 

and  meet,  [become  consecutive^  I  say,  the  angle 
rroposition.  BAD  contained  betioeen  the  chord  and  the  tan- 
gent will  be  diminished  in  finitum,  and  ultimately 

will  vanish. 

For,  if  it  does  not  vanish,  the  arc  A  CB,  will 

Demonstra-  Contain,  with  the  tangent  AD,  an  angle  equal  to 

*'°°'       a  rectilinear  angle ;  and  therefore,  the  curvature 


of 


general 


CHAP,   v.]  DIFFERENTIAL    CALCULUS. 


309 


at  the  point  A  will  not  be  continued,  which  is 
against  the  supposition. 


Lemma  VII. 

§  335.    The  same  thing  being  supposed,  I  say  statement. 
that    the   ultimate  ratic  of  the  arc,  chord,  and 
tangent,  any  one  to  any  other,  is  the  ratio  of 
equality. 

For,  while  the  point  B  approaches  towards  the  Demonstra- 
point  A,  consider  always  AB  and  AD  a^  pro- 
duced to  the  remote  points  h  and  d,  and  paral- 
lel to  the  secant  BD  draw  hd:  and  let  the  arc 
Ach  be  always  similar  to  the  axe  AC B.  Then, 
supposing  the  points 
A  and  B  to  coin- 
cide, [become  con- 
secutive], the  angle 
dAh  will  vanish,  by 
the  preceding  lem- 
ma ;  and  therefore, 
the  right  lines  Ab,  Ad  (which  were  always 
finite),  and  the  intermediate  arc  Acb,  will  co- 
incide, and  become  equal  among  themselves. 
Wherefore,  the  right  lines  AB,  AD,  and  the  conciusioa 
intermediate  arc  ACB  (wliich  are  always  pro- 
portional to  the  former),  Avill  vanish,  and  ulti- 
mately acquire  the  ratio  of  equality. 


Figrnra. 


310 


MATHEIIATICAL    SCIENCE.  [BOOK   II. 


Corollary. 


Batio. 


Ultimate 


Ratio  of 


Equality. 


Ultimate 
Batios. 


Scholium. 


Use  of 
Lemmas 


CoK.  1.  "Whence,  if  through  B  we  draw  BF 
parallel  to  the  tangent,  always  cutting  any  right 
line  AF  pass- 
ing througli  A  A  E\  \D 
and  F,  this  line  /^  Fl- 
BF  will  be,  ul- 
timately, in  the  ratio  of  equality  with  the  eva- 
nescent arc  ACB;  because,  completing  the  paral- 
lelogram AFBD,  it  is  always  in  the  ratio  of 
equality  with  AD. 

CoK.  2.  And  if  through  B  and  A  more  right 
lines  be  drawn  BE,  BD,  AF,  A  G,  cutting  the  tan- 
gent AD  and 
its  parallel  BF, 

the       ultimate  /'''^  ^l- 

ratio  of  the  ab- 
scissas AD,  AE,  BF,  BG,  and  of  the  arc  AB, 
any  one  to  any  other,  will  be  the  ratio  of 
equality. 

CoK.  3.  And  therefore,  in  any  reasoning  about 
ultimate  ratios,  we  may  freely  use  any  one  of 
those   lines  for  any  other. 

****** 

§  336.  Scholmrn. — Those  things  which  have 
been  demonstrated  of  curve  lines,  and  tlie  super- 
ficies which  they  comprehend,  may  be  easily  ap- 
plied to  the  curve  superficies,  and  contents  of 
solids.     These  lemmas  are  premised  to  avoid  the 


CHAP,  v.]         DIFFEREN-TIAL    CALCULUS.  311 

tediousness  of  deducing  perplexed  demonstrations  to  avoid  the 
ad  absurdum,  according  to  tlie  method  of  the 
ancient  geometers.  For  demonstrations  are  more 
contracted  by  the  method  of  indivisibles:  but  be- 
cause the  indivisibles  seem  somewhat  harsh,  and  use  of 
therefore,  that  method  is  reckoned  less  geometri- 
cal, I  chose  rather  to  reduce  the  demonstrations 

of  the  following  propositions  to  the  first  and  last 

-        , .  „  ,         ,  Reductio  ad 

sums  and  ratios,  ot  nascent  and  evanescent  quan- 
tities ;  that  is,  to   the  limits  of  those  sums  and 
ratios;  and  so,  to  premise,  as  sliort  as  I  could, 
the  demonstration  of  those  limits.     For,  hereby  aijgurdum 
the  same  thing  is  performed  as   by  the   method 
of  indivisibles;  and  now  those  principles  being 
demonstrated,  we  may  use  them  with  more  safety. 
Therefore,  if  hereafter  I  should  happen  to  con-    ultimate 
sider  quantities  as  made  up  of  particles,  or  should 
use  little  curve  lines  for  right  ones,  I  Avould  not 
be  understood  to  mean    indivisibles,  but  evanes-        not 
cent   divisible  quantities;   not  the  sums  and  ra- 
tios of  determinate  parts,  but  always  the  limits  of 
sums  and  ratios;  and  that  the  force  of  such  dem-  indivisible, 
onstrations  always  depends  on  the  method  laid 
down  in  the  foregoing  lemmas. 

Perhaps    it  may  be  objected,  that  there  is  no  objections 
ultimate  proportion  of  evanescent  quantities;  be- 
cause the  proportion,  before  the  quantities  have        *" 
Vanished,  is  not  the  ultimate,  and  when  they  are 


312 


MATHEMATICAL    SCIEN"CE.  [BOOK   11. 


Ultimate 
ratio ; 


Ultimate  Taiiishcd,  is  none.  But  by  the  same  argument  it 
may  be  alleged,  that  a  body  arriving  at  a  cer- 
tain place,  and  there  stopping,  has  no  nltimate 

Quantities  Velocity;  because,  the  velocity,  before  the  body 
comes  to  the  place,  is  not  its  nltimate  velocity; 
■when  it  has  arrived,  is  none.     But  the  answer  is 

answered.  G^sy ;  for,  by  the  nltimate  velocity  is  meant,  that 
with  which  the  body  is  moved,  neither  before  it 
arrives  at  its  last  place  and  the  motion  ceases, 
nor  after ;  but,  at  the  very  instant  it  arrives ; 
that  is,  that  velocity  with  which  the  body  arrives 
at  its  last  place,  and  with  which  the  motion 
ceases.  And  in  like  manner,  by  the  ultimate 
ratio  of  evanescent  quantities  is  to  be  under- 
stood the  ratio  of  the  quantities  not  before  they 
vanish,  not  afterwards,  but  with  which  they 
vanish.  In  like  manner,  the  first  ratio  of  nas- 
cent quantities  is  that  with  which  they  begin 
to  be.  And  the  first  or  last  snm,  is  that  with 
which  they  begin  to  be  (or  to  be  augmented  or 
diminished).  Tliere  is  a  limit  which  the  velocity 
at  the  end  of  the  motion  may  attain,  but  not 
exceed."  This  is  the  nltimate  velocity.  And  there 
is  the  like  limit  in  all  quantities  and  proportions 
that  begin  and  cease  to  be.  And  since  such  lim- 
its are  certain  and  definite,  to  determine  the  same 
is  a  problem  strictly  geometrical.  But  Avhatevcr 
is  geometrical  we  may  be  allowed  to  use  in  de- 


How  it  is 


to  be 


nnderstood. 


CHAP,   v.]  DIFFERENTIAL    CALCULUS. 


313 


terminmg   and   demonstrating   any   other   thing    Geometri- 
cal, 
that  is  likewise  geometrical. 


FRUITS   OF   NEWTON'S   THEORY. 

§  337.   The  main  diflSculties  in  the  higher  ma-    Newton's 

TliGorv- 

thematics,  have  arisen  from  inadequate  or  erro- 
neous notions  of  ultimate  or  evanescent  quanti- 
ties, and  of  the  ratios  of  such  quantities.  After 
two  hundred  years  of  discussion,  of  experiment 
and  of  trial,  opinions  yet  differ  Avidely  in  regard 
to  them,  and  especially  in  regard  to  the  forms 
of  language  by  Avhich  they  are  expressed. 

One  cannot  approach  this  subject,  which  has    Difficulty 
so   long    engaged   the   earnest   attention   of    the 
greatest  minds  knoAvn  to  science,  Avithout  a  feel- 
ing of  awe  and  distrust.     But  tapers  sometimes       of  the 
light  corners  which  the  rays  of  the  sun  do  not 
reach  ;  and  as  Ave  must  adopt  a  theory  in  a  sys- 
tem of  scientific   instruction,  it   is   perhaps  due     subject 
to   ethers,  that    Ave    should    assign    our   reasons 
therefor. 


§  338.   An  ultimate,    or    evanescent    quantity,    ultimate 
Avhich    is   the  basis  of  the  NeAvtonian   theory,  is      ^*° '  ^' 
not  the  quantity  '•  before  it  vanislies,  nor   after- 
wards;  but,  with  wldcli  it  vanishesP 


314  MATHEMATICAL    SCIENCE.  [BOOK   TI. 

I   have  sought,  in  what  precedes  and  follows, 
Very      to  define  this  quantity — to   separate  it  from   all 
other   quantities — to   present  it   to   the  mind  in 
Important,  a  Crystallized  form,  and  in  a  language  free  from 
all  ambiguity;   and  then   to   explain   how  it  be- 
comes the  key  of  a  sublime  science. 

As  a  first  step  in  this  process,  I  have  defined 
Firetstep    Continuous  quantity  (Art.  332),  and  this  is  the 
only  class  of  quantity  to  which  the  Differential 
Next  step.   Cal cul US  is  applicable.    The  next  step  was  to  de- 
fine consecutive  values,  and  then,  the  difi'erence 
between  any  two  of  them  (Art.  327).    These  differ- 
uitimate    ^^^ccs  are  the  ultimate   or  evanescent   quantities 
„  °^        of  Newton.      They   are  not  quantities   of  deter- 

Newton,  •'  ^ 

minate  magnitudes,  but  such  as  come  from  va- 
riables that  have  been  diminished  indefinitely.. 
They  form  a  class  of  quantities  by  themselves, 
which   have   their   own  language   and  their  own 

infinitesi-    ■'^^^^'^  ^^  change;   and  they  are  called,  Infinitesi- 
^^^^-      mals,  or  Differentials. 

Since  the  difference  between  any  two  values  of 

On  what  ^  variable  quantity,  which  are  near  together,  but 
not  consecutive,  will  depend  on  the  relative 
VALUES  of  the  quantities  and  the  law  of  change, 
it  is  plain,  tliat  when  we  pass  to  the  limit  of  this 

of  values  difference,  such  limit  will  also  depend  for  its 
value   on   the  variable  quantity  and  the   law  of 

depends,     change :   and  hence,   the   infinitesimals   are   un- 


CHAP.   V.J  DIFFEEENTIAL    CALCULUS.  315 


of 


equal  among  themselves,  and  any  two  of  them 
may  have,  the  one  to  the  other,  any  ratio  what- 
ever. 

These  infinitesimals  will  always  be  quantities  Quantities 
of  the  same  hind  as  those  from  which  they  were 
derived ;  for  the  kind  of  quantity  which  ex- 
presses a  difference,  is  the  same,  Avhether  the  dif-  same  kind, 
ference  be  great  or  small. 

LIMITS. 

§  339.    Marked    differences    of    opinion    exist     Limits. 
among   men   of    science   in   regard   to    the    true   Difference 
notion  of  a  limit;   and    hence,  definitions   have 
been  given  of  it,  differing  widely  from  each  other. 
We  have  adopted  the  views  of  Newton,  so  clearly 
set  forth  in  the  lemmas  and  scholium  which  Ave 
have    quoted  from  the  Principia.     He   uses,  as  How  defined 
stated  in  tlie  latter  part  of  the  scholium,  the  term 
limit,  to   designate    the   ultimate   or   evanescent 
value  of  a  variable  quantity;  and  this  value  is 

by  Newtcn. 

reached  under  a  particular  hypothesis.  Hence, 
our  definition  (Art.  323). 

Let  us   now  refer  again  to  the  case  of  tan- 
gency. 

Let  APB  be  any  curve  whatever,  and  TPF  a     case  of 
tangent  touching  it  at  the  point  P.     Draw  any 
chord  of  the  curve,  as  PB,  and  through  P  and  B    tangency 


of 
Definitions. 


316 


MATHEIIATICAL    SCIENCE.  [BOOK   II. 


draw  the  ordi nates  PD  and  BH.     Also  draw  PC 
parallel  to  TH. 


again  con- 


CF 

Then,   -pjj=  tang. 


FPG  =  tansr.    the    angle 


Bidered. 


Secant : 


how  it 


PTH,  which  the  tangent  line  TPF  makes  witli 
the  axis  77/". 

BO 
But,  -jj-r^  =  tangent  of  the  angle  BPC. 

If  now  we  suppose  BH 
to  move  towards  PP,  the 
angle  BPC  Avill  approach 
the  angle  FPC,  which  is 
its  limit.  When  BH  be- 
comes consecutive  Avith 
PD,  BC  will  reach  its  ul- 
timate value :  and  since  by 

Lemma  VII.,  the  ultimate  ratio  of  the  arc,  chord, 
and  tangent,  any  one  to  any  other,  is  the  ratio  of 
equality,  it  follows  that  they  must  then  all  be 
equal,  each  to  each.  Under  this  hypothesis  the 
point  B  must  fall  on  the  tangent  line  TPF;  that 
is,  the  chord  and  tangent,  in  their  ultimate  state, 
have  two  points  in  common ;  hence  they  coin- 
cide ;  and  as  the  two  points  of  the  arc  are  con- 
secutive, it  must  also  coincide  with  the  chord  and 

and  when,    tan  Sfen t. 


becomes  a 


tangent ; 


This,  at  first  sight,  seems  impossible.     But  if 
impossible,  it  be  granted  that  two  points  of  a  curve  can  be 


Not 


CHAP,   v.]  DIFFEKEXTIAL     CALCULUS.  317 


consecutive  and  that  a  straight  line  can  be  drawn 
through  any  two  points,  we  have  the  solution. 
If  we  deny  that  two  points  of  the  curve  can  be 
consecutive,  we  deny  the  law  of  continuity. 

The  method  of  Leibnitz  adopted  the  simple  Method 
hypothesis  that  when  the  point  B  approached  the 
point  P,  infinitely  near,  the  lines  CF  and  CB 
become  infinitely  small,  and  that  then,  either  may 
be  taken  for  the  other;  under  which  hypothesis 
the  ratio  of  PC  to  CB,  becomes  the  ratio  of  PC  Leibnitz. 
toCP- 


WHAT  THE  LEMMAS   OF  NEWTON"  PEOVE. 

§  340.   The  first  lemma,  which  is  "the  corner-     Lemmas 
stone  and  support  of  the  entire  system,"  predi- 
cates ultimate  equality  between  any  two  quanti- 
ties  which   continually  approach   each  other   in 

value,  and  under  such  a  law  of  change,  that,  in 

of 
any   finite   time   they  shall   approach   nearer   to 

each  other  than  by   any  given   difference.     The 

common  quantity  towards  wliich  the  quantities 

separately  converge,  is  the  limit  of  each  and  both 

of  them,  and  this  limit  is  always  reached  under    Newtoa 

a  particular  supposition. 

Lemmas  IL,  IIL,  and  IV.  indicate  the  steps  by 

which  we  pass  from  discontinuous  to  continuous      what 

quantity.     They  introduce  us,  fully,  to  the  law 


318  MATHEMATICAL     SCIEXCE.  [BOOK   II. 

they  of  continuity.  They  demonstrate  the  great 
truth,  that  the  curvilinear  space  is  the  common 
limit  of  the  inscribed  and  circumscribed  paral- 
lelograms, and  that  this  limit  is  reached  under 

the  hypothesis  that  the  breadth  of  each  parallelo- 
prove. 

gram  is  infinitely  small,  and  the  number  of  them, 

infinitely  great.     Thus  we  reach  the  law  of  con- 
tinuity; and  each  parallelogram  becomes  a  con- 
Links  in    necting  link,   in  passing  from   one   consecutive 
value  to  another,  Avhen  we  regard  the  curvilinear 
area  as  a  variable.     That  there  might  be  no  mis- 
theiawof   apprehension  in  the  matter,  corollary  1,  of  Lem- 
ma III.,  affirms,  that,  "the  ultimate  sum  of  these 
evanescent  parallelograms,  will,  in  all  parts,  coin- 
Continuity,  cide  with  the  curvilinear  figure."      Corollary  4, 
also,  affirms  that,  "therefore,  these  ultimate  fig- 
ures (as  to  their  perimeters,  acE),  are  not  recti- 
linear, but   curvilinear   limits   of  rectilinear  fig- 
Common    ures:"  that  is,  the  curvilinear  area,  A  Fa  is  the 
common  limit  of  the  inscribed  and  circumscribed 
parallelograms,   and  the   curve  Edcha,  the   com- 
Umit.      ^lon   limit  of  their  perimeters.     This   can  .only 
take  place  when  the   ordinates,  like  Dil,  Cc,  Bh, 
become  consecutive ;   and  then,  the  points  o,  n, 
m  and  I  fall  on  the  curve. 

The  law  of  continuity  carries  with  it,  neces- 

Whatthe    sarily,  the  ideas  of  the  infinitely  small  and   the 

law  of     Infinitely  great.     These  are  correlative  ideas,  and 


CHAP.   Y.]  DIFFERENTIAL    CALCULUS.  319 

in   regard  to  quantity,  one  is  the  reciprocal  of    continuity 
the  other.      The   inch   of  space,  as  well   as   the 
curved   line,  or  the   curvilinear   surface  of  geo-      i™piie«. 
metry,  has  Avithin  it  the  seminal  principles  of 
this  law. 

If  Ave  regard  it  as  a  continuous  quantity,  hav-  continuity, 
ing  increased  from  one  extremity  to  the  other, 
Avithout  missing  any  point  of  space,  we  have,  the 
laAv  of  change,  the  infinitely  small  (the  difference 
between  tAVO  consecutive  values,  or  the  link  in 
the  law  of  continuity),  and  the  infinitely  great, 
in  the  number  of  those  values  Avliich  make  up  the 
entire  line. 

It  has  been  urged  against  the  demonstrations  objections, 
of  the  lemmas,  that  a  mere  inspection  of  the  fig- 
ures proves  the  demonstrations  to  be  Avrong. 
For,  say  the  objectors,  there  Avill  be,  always,  ob- 
viously, "a  portion  of  the  exterior  parallelograms 
lying  Avithout  the  curvilinear  space."  This  is 
certainly  true  for  3injfi?iite  number  of  parallelo- 
grams. 

But  the  demonstrations   are   made  under  the  Objections 

express  hypothesis,  that,  "the  breadth  of    these 

parallelograms  be  supposed  to  be  diminished,  and 

fully 
their    number    to  be    augmented,    in  finitum." 

Under   this   supposition,   as    we    have   seen,   the 

points,  0,  n,  w,  and  /,  fall  in  the  curve,  and  then   answered. 

the  areas  named  are  certainly  equal. 


320 


MATHEMATICAL     SCIENCE.  [iJOOK   11. 


Newton's  method  in   haemony  with 
that  of  leibnitz. 

§  341.   The  method  of  treating  the  Infinitesi- 
Harmony    Dial  Calculus,  by  Leibnitz,  subsequently  ampli- 
of        fied  and  developed  by  the  Marquis  L'Hopital,  is 
based  on  two  fundamental  propositions,  or   de- 
mands, which  were  assumed  as  axioms. 

I.  That  if  an  infinitesimal  be  added  to,  or  sub- 
tracted from,  a  finite  quantity,  the  sum  or  difier- 
ence  will  be  the  same  as  the  quantity  itself  This 
demand  assumes  that  the  infinitesimal  is  so  small 
that  it  cannot  be  expressed  by  numbers. 

II.  That  a  curved  line  may  be  considered  as 
made  up  of  an  infinite  number  of  straight  lines, 
each  one  of  which  is  infinitely  small. 

It  is  proved  in  Lemma  II.  that  the  sum  of  the 
What  the  ultimate  rectangles  ^  J,  Be,  Cd,  Do,  etc.,  will  be 
equal  to  the  curvilinear  area  AaE.  This  can  only 
be  the  case  when  each  is  "less  than  any  given 
space,"  and  their  number  infinite.  What  is 
meant  by  the  phrase,  "becomes  less  than  any 
given  space  "  ?  Certainly,  a  space  too  small  to  be 
expressed  by  numbers ;  for,  if  we  have  such  a 
space,  so  expressed,  we  can  diminish  it  by  dimin- 
ishing the  number,  which  would  be  contrary  to 
Ultimate  ^^^  hypothesis.  This  ultimate  value,  then,  of 
either  of  the  rectangles,  is  numerically  zero:  and 


Methods. 


First 


demand. 


Second. 


Lemma 


provee. 


OHAP.    Y.}  DIFFEEENTIAL    CALCULUS. 


321 


Rectangles, 


hence,  its  addition  to,  or  subtraction  from,  any  value  or 
finite  quantity,  would  not  change  the  value.  The 
ultiniates  of  Newton,  therefore,  conform  to  the 
first  demand  of  Leibnitz,  as  indeed  they  should 
do ;  for,  they  are  not  numerical  quantities,  but 
connecting  links  in  the  law  of  continuity. 

It  is  proved  in  Lemma  VIL,  that  the  ultimate  Eatio 
ratio  of  the  arc,  chord,  and  tangent,  any  one  to 
any  other,  is  the  ratio  of  equality:  hence,  their  ul- 
timate values  are  equal.  When  this  takes  place,  of  chord, 
the  two  extremities  of  the  chord  become  consecu- 
tive, and  the  remote  extremity  of  the  tangent 
falls  on  the  curve,  and  coincides  Avith  the  remote 
extremity  of  the  chord :  that  is,  F  falls  on  the 
curve,  and  PB  and  PF,  coincide  with  each  other, 
and  with  the  curve.  The  length  of  this  arc, 
chord,  or  tangent,  in  their  ultimate  state,  is 


tangent, 


and 


arc,  eqnal. 


a  value  familiar  to  the  most  superficial  student  of 
the  Calculus. 

Behold,  then,  one  side  of  the  inscribed  polygon,- 
when  such  side  is  infinitely  small,  and  the  num- 
ber of  them  infinitely  great. 

That  such  quantities  as  we  have   considered, 

have    a    conceivable    existence    as    subjects    of 

thought,  and  do  or  may  have,  proximatively,  an 

actual  existence,  is  clearly  stated  in  the  latter 

21 


Value  of 
chord. 


Coinci- 
dence. 


Quantities 
have  a  real 


323  MATHEMATICAL     SCTElfCE.  [BOOK   11. 

part  of  the  scholium  quoted  from  the  Principia. 
value,      It  is  there  affirmed :  "  This  is  the  ultimate  velo- 
city.    And  there  is  a  like  limit  in  all  quantities 
and   proportions   which   begin   and   cease  to   be. 
And   since  snch  limits  are  certain  and  definite, 
ultimately,  to  determine  the  same  is  a  problem  strictly  geo- 
metrical.     But  whatever  is  geometrical  we  may 
be  allowed  to  use  in  determining   and   demon- 
strating any  other  thing  that  is  likewise  goome- 
Newton     trical."  *     Hence,  the  theory  of  Newton  conforms 

and 

Leibnitz,    to  the  second  demand  in  the  theory  of  Leibnitz. 


DIFFEEENT   DEFINITIONS   OF   A   LIMIT. 

§  342.  The  common  impression   that  mathe- 
Different    matics  is  an  exact   science,  founded  on  axioms 
of  limits,    too  obvious  to  be  disputed,  and  carried  forward 
by  a  logic   too   luminous   to   admit  of  error,  is 
certainly  erroneous  in  regard  to  the  Infinitesimal 
Calculus.     From  its  very  birth,  about  two  hun- 
dred years  ago,  to  the  present  time,  there  have 
Different    been  Very  great  differences  of  opinion  among  the 
Calculus.    ^^^^   informed  and  acutest  minds  of  each  gen- 
eration, both  in  regard  to  its  fundamental  prin- 
ciples and  to  the  forms  of  logic  to  be  employed 
in  their  development.     The  conflicting  opinions 


•  Note.  -  Tl'«  italics  are  added ;  they  are  not  in  the  text. 


CHAP,   v.]  DIFFEREKTIAL    CALCULUS.  323 

appear,  at  last,  to  have  arranged  themselves  into 
two  classes ;  and  these  differ,  mainly,  on  this 
question:  What  is  the  correct  apprehension  and  Differences, 
right  definition  of  the  word  limit?  All  seem  to 
agree  that  the  methods  of  treating  the  Calculus 
must  be  governed  by  a  right  interpretation  of 
this  word.  The  two  definitions  which  involve 
this  conflict  of  opinion,  are  these : 

1.  The  limit  of  a  variahle  quantity  is  a  qnan-      Limit 
tity  towards  which  it  may  he  made  to  approach 
nearer   than   any  given   quantity  and  ivhich  it 
reaches  under  a  particular  supposition. 

And  the  following  definition,  from  a  work  on 
the    Infinitesimal    Calculus   by   M.    Duhamel,   aji.  DuhameJ. 
French  author  of  recent  date: 

2,  The  limit  of  a  variable  is  the  constant  quan-    adCefini- 
tity  tuhich  the   variahle    indefinitely   approaches, 

hut  never  reaches. 

This  definition  finds  its  necessary  complement 

in  the  following  definition  by  the  same  author : 

"We  call,"  says  he,  "an  infinitely  small  quan-     Comple- 
ment. 
tity,  or  -simply,  an  infinitesimal,  every  variahle 

magnitude  of  ivMch  the  limit  is  zeroP 

The  diflTerence  between   the  two  definitions  is    Difference 

between 

simply  this:  by  the  first,  the  variable,  ultimately,   definitions 
reaches  its   limit;  by  the  second,   it  approaches 
the  limit,  but  never  reaches  it.    This  apparently 
sliffht  difference  in  the  definitions,  is  the  divid- 


324  MATHEMATICAL    SCIENCE.  [BOOK  II. 

ing  line  between  classes   of  profound  thinkers ; 
and  whoever  writes  a  Calculus   or  attempts  to 

Difference,  teach  the  Subject,  must  adopt  one  or  the  other  of 
these  theories.  The  first  is  in  harmony  with  the 
theories  of  Leibnitz  and  Newton,  which  do  not 
differ  from  each  other  in  any  important  particu- 
Generai  lar.  It  secms  also  to  be  in  harmony  with  the 
great  laws  of  quantity.  In  discontinous  quan- 
tity, especially,  we  certainly  include  the  limits 
in  our  thoughts,   and  in  the  forms  of  our   lan- 

Whatwe  gnage.  When  Ave  speak  of  the  quadrant  of  a 
"ttiem/  circle,  we  include  the  arc  zero  and  the  arc  of 
ninety  degrees.  Of  its  functions,  the  limits  of 
the  sine,  are  zero  and  radius;  zero  for  the  arc 
zero,  and  radius  for  the  arc  of  ninety  degrees. 
For  the  tangents,  the  limits  are  zero  and  infinity; 
zero  for  the  arc  zero,  and  infinity  for  the  arc 
of  ninety  degrees  ;  and  similarly,  for  all  the  other 
For  all      functions.     For  all  numbers,  the  limits  are  zero 

^naiititie?.  ^^^  infinity ;  and  for  all  algebraic  quantities, 
minus  infinity  and  plus  infinity. 

When   we    consider    continuous    quantity,    we 

Forcontin-  find  the  sccond  definition  in  direct  conflict  with 

"''"my"*"*  the  first  Lemma  of  Newtoti,  which  has  been 
well  called,  "  the  corner-stone  and  foundation 
of  the  Pri7icipia"  It  is  very  difficult  to  com- 
prehend that  two  quantities  may  approach  each 
other  in  value,  and  in  any  given  time  become 


CHAP,  v.]  DIFFEREKTIAL    CALCULUS. 


325 


nearer  equal  than  any  given  quantity,  and  yet 
never  become  equal ;  not  even  when  the  approacli 
can  be  continued  to  infinity,  and  when  the  law 
of  change  imposes  no  limit  to  the  decrease  of 
their  difierence.  This,  certainly,  is  contrary  to 
the  theory  of  Newton. 

Take,  for  example,  the  tangent  line  to  a  curve, 
at  a  given  point,  and  through  the  point  of  tan- 
gency  draw  any  secant,  intersecting  the  curve,  in 
a  second  point.  If  now,  the  second  point  be 
made  to  approacli  the  point  of  tangency,  both 
definitions  recognize  the  angle  Avhich  the  tangent 
line  makes  with  the  axis  of  abscissas  as  the  limit 
of  the  angles  which  the  secants  make  with  the 
same  axis,  as  the  second  point  of  secancy  ap- 
proaches the  tangent  point.  By  the  first  defini- 
tion, the  supposition  of  consecutive  points  causes 
the  secant  line  to  coincide  with,  and  become  the 
tangent.  But  by  the  second  definition,  the  se- 
cant line  can  never  become  the  tangent,  though 
it  may  approach  to  it  as  near  as  we  please.  This 
is  in  contradiction  to  all  the  analytical  methods 
of  determining  the  equations  of  tangent  lines  to 
curves.  See  corollaries  1,  2,  3,  and  4  of  Lemma 
III.,  in  which  all  the  quantities  referred  to  are 
supposed  to  reach  their  limits. 

By  the  second  definition,  there  Avould  seem  to 
be  an  impassable  barrier  placed  between  a  vari- 


In  conflict 


with 


Newton. 


Example 


of  the 


tanorent  Una 


First 


definition 


conti-ary 


to  general 


method. 


Second 
definition : 


326  MATHEMATICAL    SCIEISTCE.  [BOOK   II. 

what  it  able  quantity  and  its  limit.  If  these  two  quan- 
tities are  thus  to  be  forever  separated,  how  can 
they  be  brought  under  the  dominion  of  a  corn- 
does,  mon  law,  and  enter  together  into  the  same  equa- 
tion ?  And  if  they  cannot,  how  can  any  prop- 
erty of  the  one  be  used  to  establish  a  property  of 
the  other  ?  The  mere  fact  of  approach,  though 
Result,  infinitely  near,  would  not  seem  to  furnish  the 
necessary  conditions. 

The  difficulty  of  treating   the  subject  in  this 
Difficulty,   way  is  strikingly  manifested  in  the  supplement- 
ary definition  of  an  infinitesimal.     It  is   defined, 
simply,  as  "  every  variable  magnitude  tvhose  limit 
is  zero." 

Now,  may  not  25ero  be  a  limit  of  every  variable 
Not  which  has  not  a  special  law  of  change?  Is  not 
this  definition  too  general  to  give  a  defhstite 
idea  of  the  individual  thing  defined — an  infini- 
tesimal ?  We  have  no  crystallized  notions  of  a 
class,  till  we  apprehend,  distinctly,  the  individu- 
Shouidbe.  als  of  the  class — their  marked  characteristics — 
their  harmonies  and  their  differences;  and  also, 
their  laws  of  relation  and  connection. 

Having  given  and  illustrated  these  definitions, 

M.  Duha-    M.  Duhamel  explains  the  methods  by  which  we 

""odsnot  '  ^^"  P^^^  from  the  infinitesimals  to  their  limits; 

eatisfactory.  ^^^^^  when,  and  under  what  circumstances,  those 

limits  may  be  substituted  and  used  for  the  quan- 


CHAP.   Y.]  DIFFEKEKTIAL    CALCULUS.  327 


titles  themselves.  Those  methods  have  not 
seemed  to  me  as  clear  and  practical  as  those  of 
Newton  and  Leibnitz. 

It  is  essential  to  the  nnity  of  mathematical  Unity  in 
science,  that  all  the  definitions,  should,  as  far  as 
possible,  harmonize  witli  each  other.  In  all  dis- 
continuous quantities,  the  boundaries  are  in-  Mathematics 
eluded,  and  are  the  proper  limits.  In  the  hyper- 
bola, for  example,  we  say  that  the  asymptote  is 
the  limit  of  all  tangent  lines  to  the  curve.  But 
the  asymptote  is  the  tangent,  when  the  point  of 
contact  is  at  an  infinite  distance  from  the  ver- 

necessary. 

tex :  and  any  tangent  will  become  the  asymptote, 
under  that  hypothesis. 

If  vS  denotes  any  portion  of  a  plane  surface,  y  Differential, 
the   ordinate  and  x  the  abscissa,   we  have   the 
known  formula : 

ds  =  ydx. 

If  we   integrate   between  the  limits  of  a;  =  0, 
and  X  =  a,  we  have,  by   the    language  of  the     gm^^ce. 
Calculus 

a 


Ids  =J  ydx, 


which  is  read,  "integral  of  the  surface  between   now  read, 
limits  of  a;  =  0,  and  a;  =  a,"  in  which  both  bound- 
aries enter  into  the  result. 


328  MATHE^HATICAL    SCIENCE.  [BOOK   11. 

Limits         The    area,    actually    obtained,    begins    where 

of 

Area.       X  =  0,  and  terminates  where  x  =  a,  and   not  at 
values  infinitely  near  those  limits. 

WHAT    QUANTITIES    AKE    DENOTED    BY   0. 

§  343.   Our  acquaintance  with  the  character  0, 
whatquan-  begins  in  Arithmetic,  where  it  is  used  as  a  ne- 
cessary element  of  the  arithmetical  language,  and 
utiesare    where  it  is  entirely  without  value,  meaning,  abso- 
lutely nothing.     Used  in  this  sense,  the  largest 

denoted      ^ 

finite  number  multiplied  by  it,  gives  a  product 
^  ^        equal  to  zero ;  and  the  smallest  finite  number  di- 
vided by  it,  gives  a  quotient  of  infinity. 

When  we  come  to  consider  variable  and  con- 
May  not    tinuous  quantity,  the  infinitesimal,  or  element  of 
change  from  one  consecutive  value  to  another,  is 
be  the  Oof  ^ot  the  zero  of  Arithmetic,  though  it  is  smaller 
than    any   number   which    can    be    expressed  in 
terms  of  one,  the  base  of  the  arithmetical  system. 
New       Hence,  the  necessity  of  a  new  language.     If  the 
language    Variable  is  denoted  by  x,  we  express  the  infini- 
tesimal by  dx;  if  by  y,  then  hjdy;  and  similarly, 
for  other  variables. 

Now,  the  expressions  dx  and  dy,  have  no  exact 
How  it  is    synonyms  in  the  language  of  numbers.    As  com- 
pared with  the  unit  1,  neither  of  them  can  be  ex- 
framed,     pressed  by  the  smallest  finite  part  of  it.     Hence, 


CHAP,   v.]  DIFPEEENTIAL    CALCULUS.  329 

when  it  becomes  necessary  to  express  sucli  quan- 
tities  in   the  language   of  number,  they  can   be 

denoted  only  by  0.     Therefore,  this  0,  besides  its     What  o 
...  means, 

first  function  in  Arithmetic,  Avliere  it  is  an  ele- 
ment of  language,  and  where  tlie  value  it  denotes 
is  absolutely  nothing,  is  used,  also,  to  denote  the 
numerical  values  of  the  infinitesimals.  Hence,  it 
is  correctly  defined  as  a  character  which  some-  Sometimes 
times  denotes  absolutely  nothing,  and  sometimes 

an 

an  infinitely  small  quantity.    "We  now  see,  clearly, 

what    appears   obscure    in    Elementary  Algebra,    infinitesi 

mal. 
that  the  quotient  of  zero  divided  by  zero,  may 

be  zero,  a  finite  quantity,  or  infinity. 


IJSrSCRIBED   AND    CIECUMSCEIBED    POLYGONS 
UNITE   ON   THE    CIECLE. 

§  344.  The  theory  of  limits,  developed  by  J^cav-    inscribed 

polygon. 

ton,  is  not  only  the  foundation  of  the  higher 
mathematics,  but  indicates  the  methods  of  using 
the  Infinitesimal  Calculus  in  the  elementary 
branches.  This  Calculus  being  unknown  to  the 
ancients,  their  Geometry  was  encumbered  by  the 
tedious  methods  of  the  rechictio  ad  absurdum. 
NcAvton  says  in  the  scholium:    "These  lemmas  k  avoids  the 

reductio  ad 

are  premised   to   avoid  the  tediousness  of  dedu-   absurdnm, 
cing  perplexed  demonstrations  ad  ahsurdum,  ac- 
cording to  the  method  of  the  ancient  geometers." 


330 


MATHEMATICAL     SCIENCE.  [BOOK   II. 


Lemma  I. 


Lemma  I.,  which  is  the  "  corner-stone  and 
foundation  of  the  Principia^  is  also  the  golden 
link  which  connects  geometry  with  the  higher 
mathematics. 

It  is  demonstrated  in  Euclid's  Elements,  and 
also  in  Davies'  Legendre,  Book  V.,  Proposition 
X.,  that  '-'  T'wo  regular  ijolygons  of  ilie  same  num- 
ber of  sides  can  he  constriicted,  tlie  one  circum- 
scribed about  the  circle  and  the  other  inscribed 
witliin  it,  tchich  shall  differ  from  each  other  by 
etrated.     i^gg  t]ian  ciuy  given  surface." 

The   nfoment   it    is   proved  that   the   exterior 


What  is 


demon- 


What 
Newton 
affirms. 


How  the  and  interior  polygons  may  be  made  to  differ 
made.  from  each  other  by  less  than  any  given  surface, 
Lemma  L  steps  in  and  affirms  an  ultimate  equal- 
ity between  them.  And  when  does  that  ultimate 
equality  take  place,  and  when  and  where  do  they 
become  coincident  ?  Newton,  in  substance  af- 
firms, in  his  lemmas,  "on  their  common  limit, 
the  circle,"  and  under  the  same  hypothesis  as 
causes  the  inscribed  and  circumscribed  parallelo- 
grams to  become  equal  to  their  common  limits, 
the  curvilinear  area.  If  Lemma  I.  is  true,  the 
perimeters  of  the  two  polygons  will  ultimately 
coincide  on  the  circumference  of  the.  circle,  and 
A  side  of  the  become  equal  to  it.  But  what  then  is  the  side  of 
'^^^°°"  each  polygon?  We  answer,  the  distance  between 
two  consecutive  points  of  the  circumference  of 


CHAP,   v.]         DIFFERENTIAL    CALCULUS.  331 


the  circle.    And  what  is  that  value  ?    We  answer, 
the  V  dx^  +  dif. 

But  it  is  objected,  that  this  introduces  us  to  objectionB 
the  infinitely  small.  True,  it  does;  but  we  can- 
not reach  a  continuous  quantity  without  it.  The 
sides  of  the  polygons,  so  long  as  their  member  ^^  ^.^^^^ 
is  finite,  will  be  straight  lines,  each  diminishing 
in  value  as  their  number  is  increased.  While 
this  is  so,  the  perimeter  of  each  will  be  a  discon- 

discussed 

tinuous    quantity,  made   up  of  the   equal   sides, 
each  having  a  finite  value,   and  each  being  the 
unit  of  change,  as  we  go  around  the  perimeter.        an^ 
As  each  of  these  sides   is   diminished   in  value, 
and   their   number  increased,  the  discontinuous 
quantity  approaches  the  law  of  continuity,  which  considered, 
it  reaches,  under  the  hypothesis,  that  each  side 
becomes   infinitely  small  and  their   number  in- 
finitely great.      Behold  the  polygons  embracing   where  the 
each  other  on  their  common  limit,  the  circle,  and 
the  perimeter  of  each  coinciding  Avith  the   cir-    twopoiy- 

cumference.     Thus,  the  principles  of  the  Infini- 

gons  em-< 

tesimal  Calculus  take  their  appropriate  place  m 
Elementary   Geometry,  to   the    exclusion  of  the   ^jyaceeach 
cumbrous  methods  of  the  rednctio  ad  aisurdtcm 
of  thb  ancients,  and  the  whole  science  of  Mathe-      other, 
matics  is  brought    into    closer    harmonies  and 
nearer  relations. 


332  MATHEMATICAL     SCIENCE.  [BOOK  XL 

DIFFERENTIAL   AND    INTEGRAL   CALCULUS. 

§  345.   We  have  seen  that  the  DiiSPerential  and 
Differential  Integral   Calculus   is   conversant    about   contin- 
uous  quantity.     We   have   also   seen,  that   such 
quantities    are    developed    by   considering   their 

and  Integral 

laws  of   change.      We   have    further   seen,   that 

these  laws  of  change  are  traced  by  means  of 
Calculus    ^j-^g  differences  of  consecutive  values,  taken  two 

and  two,  as  the  variables  pass  from  one  state  of 
defined,     value  to  another.     Indeed,  those  differences  are 

but  the  foot-steps  of  these  laws. 

LANGUAGE   OF  THE   CALCULUS. 

§  346.   We  are  now  to  explain  the  language  by 

Language    which  the  quantities  are  represented,  by  Avhich 

their  changes  are  indicated,  and  by  which  their 

laws  of  change  are  traced.     The  constant  quan- 
of  the  ^  ^ 

titles  which  enter  into  the  Calculus  are  repre- 
sented by  the  first  letters  of  the  alphabet,  a,  h, 
c,  etc.,  and  the  variables,  by  the  .final  letters,  x, 
y,  z,  etc. 

When   two   variable   quantities,  y  and   .t,  are 
Variable    couiiected  in  an  equation,  either  of  them   may 

quantities,    ,  -i    ,       •  i  -j-         i 

be  supposed  to  increase  or  decrease  unijormly ; 
such  variable  is  called  the  independent  variable, 
because  the  laio  of  cliange  is  arhitrary,  and  in- 


CHAP,   v.]  DIFFEKENTIAL    CALCULUS.  333 

dependent   of  the  form  of  the   equation.      This  Function  of 

each  other. 

variable  is  generally  denoted  by  x,  and  called 
simply,  the  variahle.  Under  this  hypotliesis,  the 
change  in  the  variable  y  will  depend  on  the  form 
of  the  equation :  hence,  y  is  called  the  dependent 
variable,  ov  function.  When  such  relations  exist  How  they 
between  y  and  x,  they  are  expressed  by  an  equa- 
tion of  the  form 


y  =  F  {x),     y~f  {x),  or,     /  {x,  y)  -  0, 


maybe 


which  is  read,  y  a  function  of  x.    The  letter  F^ 

or  /,  is  a  mere  symbol,  and-  stands  for  the   word   gxpreggcd, 

function,    li  y  is  a  function  of  a;,  that  is,  changes 

with  it,  X  may,  if  we  please,  be   regarded   as   a 

function  of  y ;  hence, 

One  qiiantUy  is  a  function  of  another,  wlien  the   Function. 
ttoo  are  so  connected  that  any  change  of  value,  in 
either,  produces  a  corresponding  change  in  the 
other. 

It  has  been  already  stated  (Art.  328),  that  the  Difference 
difference  between  two  consecutive  values  of  a 
variable  quantity,  is  indicated  by  simply  writing 
the  letter  (Z  as  a  symbol,  before  the  letter  denot- 
ing that  variable ;  so  that  dx  denotes  the  differ* 
ence  between  two  consecutive  values  of  the  vari- 
able quantity  denoted  by  x,  and  dy  the  difference 
between  the  corresponding  consecutive  values  of 


334  MATHEMATICAL     SCIENCE.  [BOOK   II. 

Form  of     the  Variable  quantity  denoted   by  y.     These  are 

language. 

mere  forms  of  language,  expressing  laws  of 
change. 

How  are  the  changes  in  these  variable  quan- 
standard  of  titles,  expressed  by  the  infinitesimals,  to  be  meas- 
ured ?     Only  by  taking  one  of  them  as  a  standard 
'    — and  finding  how  many  times  it  is  contained  in 
the  other. 

The  independent  variable  is  always  supposed 
Independent  to  increase  tmiformly  ;  hence,  the  difference  be- 
tween any  two  of  its  consecutive   values,  taken 
Change     at  pleasure,  is  the  same  :  therefore,  this  difference, 

nnifoi'm. 

Avhich  does  not  vary  in  the  same  equation,  or 
under  the  same  law  of  change,  affords  a  con- 
venient standard,  or  unit  of  measure,  and  in  the 
Calculus,  is  always  used  as  such. 

The  change  in  the  function  y,  denoted  by  dy, 

CoTTespond-  is     always     compared    with    the     corresponding 

change  of  the  independent  variable,  denoted  by 

dx,  as  a  standard,  or  unit  of  measure.     But  the 

change  in  any  quantity,  divided  by  the  unit  of 

measure,  gives  the  rate  of  change:  hence,  -r^  is 

in  the  '    *=  o  '  f/^ 

the  rate  of  change  of  the  function  y.     This  rate 

of  change  is  called  the  differential  coeMcient  of 
function,  °  jj  jj 

y  regarded  as  a  function  of   x,  and   performs  a 

very  important  part  in  the  Calculus.     The  quan- 

not  uniform.  ^|^|gg  j^^  ^^^^  ^^^  being  both   infinitesimals,  are 


Ing  change 


CHAP,   v.]  DIFFERENTIAL    CALCULUS.  335 

of  the  same  species :  hence,  their  quotient  is  an 
abstract  numler.  Therefore,  the  differential  co- 
efficient is  a  connecting  link  between  the  infini- 
tesimals and  numbers. 

If   any   quantity  whatever   be   divided   by  its  Quotient  by 
unit  of    measure,   the   quotient   will   be  an    ab-    measure, 
stract  number;  and  if  this  quotient  be  multiplied 
by  the  unit  of  measure,  the   product  will  be  tlie 
concrete  quantity  itself     Hence,  if  we  multiply 

-7^,  by  the  unit  of  measure  clx,  we  have  -—■  dx, 
dx     ■^  dx 

which  always  denotes  tlie  difference  between  two 
consecutive  values  of  y\  and  therefore,  is  the  dif- 
ferential of  y.     Hence,  the  differential  of  a  vari-  Differential 
aUe  function  is  equal  to  the  differential  coefficient    quantity. 
multiplied  iy  the  differential  of  the  independent 
variable. 

The  method,  therefore,  of  dealing  with  infini-       same 
tesimals,  is  precisely  the  same  as  that  employed 
for  discontinuous  quantities. 

We  assume  a  unit  of  measure  which  is  as  arbi-  unit  of 
trary  as  one,  in  numbers,  or,  as  the  foot,  yard,  or 
rod,  ill  linear  measure,  and  then  we  compare  all 
other  infinitesimals  with  this  standard.  We  thus 
obtain  a  ratio  which  is  an  abstract  number, 
and  if  this  be  multiplied  by  the  unit  of  measure, 
we  go  back  to  the  concrete  quantity  from  which 
the  ratio  was  derived. 


measure. 


336 


MATHEMATICAL    SCIEN"CE.  [BOOK  II. 


We  have  thus  sketched  an  outline  of  the  In- 
finitesimal Calculus.  We  have  named  the  quan- 
tities about  which  it  is  conversant,  the  laws 
which  govern  their  changes  of  value,  and  the 
language  by  which  these  laws  are  expressed.  We 
have  found  here,  as  in  the  other  branches  of 
mathematics,  that  an  arbitrary  quantity,  assumed 
as  a  unit  of  measure,  is  the  base  of  the  entire 
system ;  and  that  the  system  itself  is  made  up  of 
the  various  processes  employed  in  finding  the 
CaicuiQB.  ratio  of  this  standard,  to  the  quantities  which  it 
measures. 


Sketch 


of  the 


Inflnitesi 
mal 


APPENDIX. 


A   COTTHSE    OF    MATHEMATICS WHAT    IT    SHOULD    BE. 

§  347.  A  COURSE  of  mathematics  should  pre-  a  course 
sent  the  outUnes  of  the  science,  so  arranged,  ex-  Mathematics 
plained,  and  illustrated  as  to  indicate  all  those 
general  methods  of  application,  which  render  it 
effective  and  useful.  This  can  best  be  done  by 
a  series  of  works  embracing  all  the  topics,  and 
in  which  each  topic  is  separately  treated. 

§  348.   Such  a  series  should  be  formed  in  ac-  How  it 

cordance  with  a  fixed  plan  ;  should  adopt  and  formed, 
use  the  same  terms  in  all  the  branches ;  should 
be  written  throughout  in  the  same  style ;    and 

present  that  entire  unity  which  belongs  to  the  unity  of  the 

subject  itself.  ''"''^"'"• 

§  349.    The  reasonings   of  mathematics   and  Reasoninga 
the  processes  of  investigation,  are  the  same  in 

22 


APPENDIX. 


the  same  in   cvery  branch,  and  have  to  be  learned  but  once, 

if  the  same  system  be  studied  throughout.     The 

Different     different  kinds  of  notation,  though  somewhat  un- 

kiiids  of  no- 

tauou.      hke  in  the  different  subjects  of  the  science,  are, 
in  fact,  but  dialects  of  a  common  language. 


Language         §  350.    If^   then,   the  language   is,   or  may  be 

need  be 


onco. 


In  what 
consists  tlie 
difficulty  V 


learned  but  made  essentially  the  same  in  all  the  branches  of 
mathematical  science  ;  and  if  there  is,  as  has 
been  fully  shown,  no  difference  in  the  processes 
of  reasoning,  wherein  lies  that  difficulty  in  the 
acquisition  of  mathematical  knowledge  which  is 
often  experienced  by  students,  and  whence  the 
origin  of  that  opinion  that  the  subject  itself  is 
dry  and  difficult  ? 

A  §  351.  Just  in  proportion  as  a  branch  of  know- 

general  law, 

if  known,    ledge  is  compactly  united  by  a  common  law,  is 
Bubect'eM-  ^^®  facility  of  acquiring  that  knowledge,  if  we 
observe  the   law,  and  the  difficulty  of  acquiring 
Faculties     it,  if  wc  pay  no  attention  to  the  law.     The  study 
DiXemati'cs.  ^f   mathematics   demands,  at   every  step,   close 
attention,  nice  discrimination,  and  certain  judg- 
ment.     These  faculties  can  only  be  developed 
How  first    by  culture.     They  must,  like  other  faculties,  pass 
through  the  states  of  infancy,  growth,  and  ma- 
turity.    They  must  be  first  exercised  on  sensible 
and   simple   objects ;     then   on   elementaj-y    ab- 


APPENDIX.  339 


stract  ideas  ;  and  finally,  on  generalizations  and     on  what 

the  hi 

ideal. 


the  higher  combinations  of  thought  in  the  pure      cised. 


§  352.  Have  educators  fully  realized  that  the    Arithmetic 

th:;  most  inv 

first  lessons  in  numbers  impi'ess  the  first  elements      poitant 
of  mathematical    science  ?    that    the  first    con-       '^'^^  " 
nections  of  thought  which  are  there  formed  be- 
come the  first  threads  of  that  intellectual  warp 
which  gives   tone    and  strength    to    the    mind  ? 
Have  they  yet  realized  that  every  process  is,  or     ah  thB 

1         1  1  1  iM  I  n  ^        r  i  subjects  coi» 

should  be,  like  the  stone  of  an  arch,  formed  to  nected. 
fill,  in  the  entire  structure,  the  exact  place  for 
which  it  is  designed  ?  and  that  the  unity,  beauty, 
and  strength  of  the  whole  depend  on  the  adapta- 
tion of  the  parts  to  each  other  ?  Have  they 
sufficiently  reflected  on  the  confusion  which  must    Necessity 

™  .  I  1    1  of  unity  in  all 

arise  irom  attempting  to  put  together  and  har-    the  parts, 
monize  different   parts   of  discordant   systems  ? 
to  blend  portions  that  are  fragmentary,  and  to 
unite  into  a  placid  and  tranquil  stream  trains  of 
thought  which  have  not  a  common  source  ? 

§  353.   Some  have  supposed  that  Arithmetic 
may  be  well  taught  and  learned  without  the  aid 
of  a  text-book  ;  or,  if  studied  from  a  book,  that  a  text-book 
the  teacher  may  advantageously  substitute  his 
own  methods  for  those  of  the  author,  inasmuch 


340  A  P  P  E  V  D  I  X 

tobefoi-     as  such  substitution  is  calculated  to  widfen  the 
field  of  investigation,  and  excite  the  mind  of  the 
pupil  to  new  inquiries. 
Ecasons.  Adnfiitting  that  every  teacher  of  reasonable 

intelligence,  will  discover  methods  of  communi- 
cating instruction  better  adapted  to  the  peculiar- 
ities of  his  own  mind,  than  all  the  methods  em- 
Kven  a  bet-   ployed  by  the  author  he  may  use  ;  will  it  be  safe, 

tcr  method, 

when  8ubsti-  as  a  general  rule,  to  substitute  extemporaneous 
not^h'amj^  mcthods  for  those  which  have  been  subjected 
nize  with  the  ^^  ^j^^  analysis  of  science  and  the  tests  of  expe- 

other  parta  ''  '■ 

of  the  work,  rjencc  ?  Is  it  safe  to  substitute  the  results  of 
conjectural  judgments  for  known  laws?  But  if 
they  are  as  good,  or  better  even,  as  isolated  pro- 
cesses, will  they  answer  as  well,  in  their  new 
places  and  connections,  as  the  parts  rejected? 

Illustration.  Will  the  balance-whcel  of  a  chronometer  give 
as  steady  a  motion  to  a  common  watch  as  the 
more  simple  and  less  perfect  contrivance  to 
which  all  the  other  parts  are  adapted  ? 

§  354.  If  these  questions  have  significance,  we 

One  of  the    have  fouud  at  least  one  of  the  causes  that  have 

mathematics  impeded  the  advancement  of  mathematical  sci- 

fe  (i.fflcuit.    QT^QQ^  yJ2.  the  attempt  to  unite  in  the  same  course 

of  instruction  fragments   of  difterent   systems ; 

thus  presenting  to  the  mind  of  the  learner  the 

same    terms  differently    defined,    ar.d    the    same 


APPENDIX^  341 


principles  differently  explained,    illustrated,    and 
applied.      It    is   mutual  relation  and   connection   Connection 

very  impor- 

which  bring  sets  of  facts  under  general  laws  ;  it       tant. 
is  mutual  relation  and  connection  of  ideas  which 
form  a  process  of  science ;  it  is  the  mutual  con- 
nection   and   relation  of  such   processes  which 
constitute  science  itself 


§  355.  I  would  by  no  means  be  understood  as    ^  teacher 
expressina;  the  opinion  that  a  student  or  teacher  ^'^""id jead 

i^  ^  ^  many  booka 

of  mathematics  should  limit  his  researches  to  a  and  teach  one 

system. 

single  author ;  for,  he  must  necessarily  read  and 
study  many.  I  speak  of  the  pupil  alone,  who 
must  he  taught  one  method  at  a  time,  and  taught 
that  well,  before  he  is  able  to  compare  different 
methods  with  each  other. 


ORDER  OF  THE  SUBJECTS ARITHMETIC. 

§  356.    Arithmetic   is    the    most   useful   and  Arithmetic 
simple  branch  of  mathematical  science,  and  is 
the  first  to  be   taught.     If,  however,  the  pupil 
has  time  for  a  full  course,  I  would  by  no  means   connection 
recommend  him  to  finish  his  Arithmetic  before     Algebra. 
studying  a  portion  of  Algebra. 


343  APPENDIX. 


ALGEBRA. 

Algebra:  §  ^^'^'  Algebra  is  but  a  universal  Arithmetic; 
with  a  more  comprehensive  notation.  Its  ele- 
ments are  acquired  more  readily  than  the  higher 
and  hidden  properties  of  numbers ;  and  indeed 
the  elements  of  any  branch  of  mathematics  are 
more  simple  than  the  higher  principles  of  the 
How       preceding  subject ;  so  that  all  the   subjects  can 

O  should  be 

studied:     bcst  be  Studied  in  connection  with  those  which 
preceue  and  follow. 

Should  §  358.  Algebra,  in  a  regular  course  of  instruc- 

pfgcgcIb 

Geometry:   tiou,  should  prcccdc  Gcomctry,  because  the  ele- 
mentary processes  do  not  require,  in  so  high  a 
Why.       degree,  the  exercise  of  the  faculties  of  abstrac- 
tion   and   generalization.      But   when    we    have 
When      completed  the   equation    of  the    second  degree, 
should  be    the  proccsses  become  more  difficult,  the  abstrac- 
commenced.  ^.j^j^g  jj^gre  perfect,  and  the  generalizations  more 
extended.     Here  then  I  would  pause  and  com- 
mence Geometry. 

GEOMETRY. 

Geometry,  §  359.  Geometry,  as  one  of  the  subjects  of 
mathematical  science,  has  been  fully  considered 
in  Book  II.  It  is  referred  to  here  merely  to  mark 
its  place  in  a  regular  course  of  instruction. 


APPENDIX.  343 


TRIGONOMETRY PLANE    AND    SPHERICAL. 

§  360.  The  next  subject  in  order,  after  Geom-   Trigonome. 

try: 

etry,  is  Trigonometry :  a  mere  application  of  the 
principles  of  Arithmetic,  Algebra,  and  Geometry    what  u  li. 
to  the  determination  of  the  sides  and  angles  of 
triangles.     As   triangles    are   of  two  kinds,   viz. 
those  formed  by  straight  lines  and  those  formed 
by  the  arcs  of  great  circles  on  the   surface  of  a 
sphere ;    so  Trigonometry  is   divided    into  two  Two  kin.K 
parts :  Plane  and   Spherical.     Plane    Trigonom- 
etry explains  the   methods,   and  lays   down  the      P'^ne. 
necessary  rules  for  finding   the  remaining  sides 
and  angles  of  a  plane  triangle,  when  a  sufficient 
number  are  known  or  given.     Spherical  Trigo-    sphericau 
iiometry  explains  like  processes,  and  lays  down 
similar  rules  for  spherical  triangles. 


SURVEYING  AND  LEVELLING. 

§  361.  The  application  of  the  principles  of 
Trigonometry  to  the  measurement  of  portions 
of  the  earth's  surface,  is  called  Surveying;  and  surveying 
similar  applications  of  the  same  principles  to  the 
determination  of  the  difference  between  the  dis- 
tances of  any  two  points  from  the  centre  of  the 
earth,  is  called  Levelling.  These  subjects,  which  levelling, 
tbliow  Trigonometry,  not  only  embrace  the  va- 


344  APPENDIX. 


What  they  rious  mcthods  of  calculatioii,  but  also  a  descrip- 
tion of  the  necessary  Instruments  and  Tables. 
They  should  be  studied  immediately  after  Trigo- 
nometry ;  of  which,  indeed,  they  are  but  appli- 
cations. 

DESCRIPTIVE     GEOBIETRY. 

Descriptive        §  362.    Descriptivc   Geometry  is   that  branch 

Geometry: 

of  mathematics  which  considers  the  positions  of 
the  geometrical  magnitudes,  as  they  may  exist  in 
space,  and  determines  these  positions  by  refer- 
ring the  magnitudes  to  two  planes  called  the 
Planes  of  Projection. 

Its  nature.  It  is,  indeed,  but  a  development  of  those  gen- 
eral methods,  by  which  lines,  surfaces,  and  vol- 
umes may  be  presented  to  the  mind  by  means 
of  drawings  made  upon  paper.  The  processes  of 
What  it3     this  development  require  the  constant  exercise  of 

**"  Lhei"*  ^^®  conceptive  faculty.  All  geometrical  mag- 
nitudes may  be  referred  to  two  planes  of  pro- 
jection, and  their  representations  on  these  planes 
will  express  to  the  mind,  their  forms,  extent,  and 
also  their  positions  or  places  in  space.  From 
How.  these  representations,  the  mind  perceives,  as  it 
were,  at  a  single  view,  the  magnitudes  them- 
selves, as  they  exist  in  space  ;  traces  their  boun- 
daries, measures  their  extent,  and  sees  all  their 
parts  separately  and  in  their  connection. 


APPENDIX.  34o 

In  France,  Descriptive  Geometry  is  an  irnpor-       how 

1  r-       1  •  T     •  1       •  regarded  ia 

tant  element  ot  education.  It  is  taught  in  most  France, 
of  the  public  schools,  and  is  regarded  as  indis- 
pensable to  the  architect  and  engineer.  It  is, 
indeed,  the  only  means  of  so  reducing  to  paper, 
and  presenting  at  a  single  view,  all  the  compli- 
cated parts  of  a  structure,  that  the  drawing  or 
representation  of  it  can  be  read  at  a  glance,  and 
all  the  parts  be  at  once  referred  to  their  appropri- 
ate places.     It  is  to  the  engineer  or  architect  not     its  value 

1  .1  11-11  ,   as  a  practicai 

only  a  general  language  by  which  he  can  record      branch. 
and  express   to   others  all  his  conceptions,  but  is 
also  the  most  powerful  means  of  extending  those 
conceptions,  and  subjecting  them  to  the  laws  of 
exact  science. 


SHADES,    SHADOWS,    AND    PERSPECTIVE. 

§  3G3.  The  application  of  Descriptive  Geom- 
etry to  the  determination  of  shades  and  shadows,      shades, 
^  as  they  are  found  to   exist  on  the  surfaces  of       and 
bodies,  is  one  of  the  most  striking  and  useful  ap-  ^^''^p'^'^'^^* 
plications  of   science  ;    and  when   it    is   further 
extended  to  the  subject  of  Perspective,  we  have 
all  that  is  necessary  to  the  exact  representation 
of  objects  as  they  appear  in  nature.     An  accu- 
rate perspective   and   the   right   distribution   of 
light  and  shade  are  the  basis  of  every  work  of 


J46 


APPENDIX. 


Thoiruse.  the  fine  arts.  Without  them,  the  sculptor  and 
the  painter  would  labor  in  vain :  the  chisel  of 
Canova  would  give  no  life  to  the  marble,  nor  the 
touches  of  Raphael  to  the  canvas. 


ANALYTICAL     GEOMETRY, 


Analytical 
Geometry. 


Its 
Importance 


a  «t"dy. 


§  364.  Analytical  Geometry  is  the  next  sub- 
ject in  a  regular  course  of  mathematical  study, 
though  it  may  be  studied  before  Descriptive  Ge- 
ometry. The  importance  of  this  subject  cannot 
be  exaggerated.  In  Algebra,  the  symbols  of 
quantity  have  generally  so  close  a  connection 
with  numbers,  that  the  mind  scarcely  realizes 
Valuable  as  the  extent  of  the  generalization ;  and  the  power 
of  analysis,  arising  from  the  changes  that  may- 
take  place  among  the  quantities  which  the  sym- 
bols represent,  cannot  be  fully  explained  and  de- 
veloped. 

But  in  Analytical  Geometry,  where  all  the 
magnitudes  are  brought  under  the  power  of  anal- 
ysis, and  all  their  properties  developed  by  the 
combined  processes  of  Algebra  and  Geometry,  we 
are  brought  to  feel   the  extent  and  potency  of 

Qeiicraiiza-  those  methods  which  combine  in  a  single  equa- 
tion. 

tion  every  discovered  and  undiscovered  property 

of  every  line,  straight  or  curved,  which  can  be 

formed  by  the  intersection  of  a  cone  and  plane. 


Reasons. 


APPENDIX.  347 


To  develop  every  property  of  the  Conic  Sec-  lu  extent. 
tions  from  a  single  equation,  and  that  an  equa- 
tion only  of  the  second  degree,  by  the  known 
processes  of  Algebra,  and  thus  interpret  the  re- 
sults, is  a  far  different  exercise  of  the  mind  from 
that  which  arises  from  searching  them  out  by 
the  tedious  and  disconnected  methods  of  separate 
propositions.  The  first  traces  all  from  an  inex-  iia  methods 
haustible  fountain,  by  the  known  laws  of  analyti- 
cal investigation,  applicable  to  all  similar  cases, 
while  the  latter  adopts  particular  processes  ap- 
plicable to  special  cases  only,  without  any  gen- 
eral law  of  connection. 


DIFFERENTIAL    AND    INTEGRAL    CALCULUS. 

§  365.  The  Differential  and  Integral  Calculus  Differeutiaj 

presents  a  new  view  of  the  power,  extent,  and  imec^rai 

applications  of  mathematical  science.     It  should  caiciuus. 

be  carefully  studied   by  all  who  seek  to  make  what  per- 

,  .    ,  .  .  ,  •       1    1  11  sons  should 

high  attainments  in  mathematical  knowledge,  or  study  it. 
who  desire  to  read  the  best  works  on  Natural 
and  Experimental  Philosophy.  It  is  that  field  of 
mathematical  investigation,  where  genius  may 
exert  its  highest  powers  and  find  its  most  certain 
rewards.  It  reaches,  with  a  microscopic  certain- 
ty the  most  hidden  laws  of  quantity,  and  brings 
them  within  the  range  of  Mathematical  Anal3'sis. 


348  APPENDIX. 


Continuous      Contiiiuous  Quantity,  under  all  its  forms,  and 
quan  i  y.    ^^.^^^  ^^^^  ^^^  infinite  laws  of  cliange,  can  be  ex- 
amined and  analyzed  only  by  the  Calculus. 
Language        The  language  constructed  for  the  development 
of  the   laws   and  properties  of  quantities   com- 
°^        posed  of  ascertained  and  definite  parts,  is  inap- 
plicable  to  quantity  changing  according  to  the 

discontinu-  , .       . ,  tt-  n  ^  ^ 

law  of  continuity.     Here,  the  changes  can  only 
,.,   be  expressed  by  infinitesimals,  which  are   mere 

one  quantity  L  J  ' 

links  in  the  law  of  change,  and  which  form  no 
inappiica-    appreciable  part  of  the  quantity  itself.     We  are 

ble. 

thus  introduced  to  a  new  form  of  Mathematical 
Science.      It  is   this    science   which    deals   with 

What  the  Time,  and  Space,  and  Force,  and  Motion,  and 
Velocity,  and  indeed,  with  all  Continuous  Quan- 

language  tlty.  The  elements  of  this  science  are  infinitesi- 
mal; but  the  science  itself  reaches  through  all 
deals  with,  ^jj^^g  j^^^^  ^11  space,  revealing  the  mysteries  and 
the  omnipotence  of  universal  law. 


BOOK    III. 

UTILITY    OF   MATHEMATICS. 


CHAPTER    I. 

THE   UTILITY    OF    MATHEMATICS    CONSIDEREfi    AS    A    MEANS    OF    iNXELtECTUAt 
TRAINING    AND    CULTURE. 

§366.    The  first  efibrts  in  mathematical  sci-   First  effona 
ence  are  made  by  the  child  in  the  process  of 
counting.      He  counts  his  fingers,  and  repeats 
the  words  one,  two,  three,  four,  five,  six,  seven,   conntmgof 

sensible  ol>> 

eight,  nine,  ten,  until  he  associates  with  these  jecta, 
words  the  ideas  of  one  or  more,  and  thus  ac- 
quires his  first  notions  of  number.  Hence,  the 
idea  of  number  is  first  presented  to  the  mind  by 
means  of  sensible  objects ;  but  when  once  clear- 
ly apprehended,  the  perception  of  the  sensible 
objects  fades  away,  and  the  mind  retains  only 
the  abstract  idea.     Thus,  the  child,  after  count-    GcnenHzM 

tioth 

ing  for  a  time  with  the  aid  of  his  fingers  or  his 
marbles,  dispenses  with  these  cumbrous  helps,  and 


350  UTILITY     UF     MATHEMATICS.  [uOOK  111. 

Abstraction,  employs  Only  the  abstract  ideas,  which  his  mind 
embraces  with  clearness  and  uses  with  facility. 


Analytical  §  3G7.    In  the   first   stages   of  the   analytical 

method:  i       i  i  i  •   •  •  i  i 

methods,  where  the  quantities  considered  are 
Tses  sensible  represented  by  the  letters  of  the  alphabet,  sen- 
first,  sible  objects  again  lend  their  aid  to  enable  the 
mind  to  gain  exact  and  distinct  ideas  of  the 
things  considered ;  but  no  sooner  are  these  ideas 
obtained  than  the  mind  loses  sight  of  the  things 
themselves,  and  operates  entirely  through  the 
instrumentality  of  symbols. 


Geometry.         §  3G8.  So,  also,  in  Geometry.     The  right  line 

may  first  be  presented  to  the  mind,  as  a  black 

First  impres-  mark   Oil  paper,  or  a  chalk  mark   on  a  black- 

Bions  by  sen    ,  ,  .  ,  .,,....  , 

Bible  objects.  Doard,  to  imprcss  the  geometrical  dennition,  thai 
"  A  straight  line  does  not  change  its  direction 
between  any  two  of  its  points."  When  this 
definition  is  clearly  apprehended,  the  mind  needs 
no  further  aid  from  the  eye,  for  the  image  is 
forever  imprinted. 

A  plane.         g  3g9_  T^g  ifjga  of  a  plane  surface  may  be 
Definition:    impressed  by  exhibiting  the  surface  of  a  polished 
mirror;    and    thus    the  mind    may  be    aided  in 
Bow  iiuistia-  apprehending  the  definition,  that   "  a  plane  sur- 
face is  one  in  which,  if  any  two  points  be  taken 


CFIAP.   I.] 


QUANTITY SPACE, 


351 


the  straight  Hne  which  joins  them  will  he  wholly 
in  the  surface."  But  when  the  definition  is 
understood,  the  mind  requires  no  sensible  object      it*  true 

conception. 

to  aid  its  conception.  The  ideal  alone  fills  the 
mind,  and  the  image  lives  there  without  any 
connection  with  sensible  objects. 


§  370.   Space  is  indefinite  extension,  in  which      space, 
all  bodies  are  situated.    A  volume  is  any  limited    Volume-, 
portion  of  space  embracing  the  three  dimensions 
of  length,  breadth,  and  thickness.     To  give  to  the 
mind  the  true  conception  of  a  volume,  the  aid    How  con. 

ccivcd. 

of  the  eye  may  at  first  be  necessary;  but  the 
idea  being  once  impressed,  that  a  volume,  in  a 
strictly  mathematical  sense,  means  only  a  por- 
tion of  space,  and  has  no  reference  to  the  mat- 
ter with  which  the  space  may  be  filled,  the  mind 
turns  away  from  the  material  object,  and  dwells 
alone  on  the  ideal. 


Wliat  It 
really  is. 


§  371.  Although  quantity,  in  its  general  sense,     Quantity: 

is  the  subject  of  mathematical  inquiry,  yet  the 

anguage  of  mathematics  is  so  constructed,  that    language; 

the  investigations  ai'e  pursued  without  the  slight-    How  con- 
structed. 
est  reference  to  quantity  as  a  material  substance. 

We   have   seen   that   a   system    of   symbols,   by 

which  quantities  may  be  represented,  has   been    symboisi 

adopted,  forming  a  language  for  the  expression 


852  UTILITY     OF     MATHEMATICS.  [boOK  III. 

of  ideas  entirely  disconnected  from  material  ob- 
jects, and  yet  capable   of  expressing  and  repre- 
Nature  of    scntiiig  such  objects.     This  symbolical  language, 

the  lan- 
guage:     at  once  copious  and  exact,  not  only  enables  us 

to  express  our  known  thoughts,  in  every  depart- 

whatitao-  mcut  of  mathematical  science,  but  is   a  potent 

compliahes.  ^  ,  .  .  .    .         .     ^  ,  ■, 

means  oi  pushing  our  mquu'ies  into  unexplored 
regions,  and  conducting  the  mind  with  certainty 
to  new  and  valuable  truths. 

Advantages       §  ^'^^-  The  nature  of  that  culture,  which  the 
of  an  exact   j^jj^^j  undergoes  by  being  trained  in  the  use  of 

language.  o  ./  o 

an  exact  language,  in  which  the  connection  be- 
tween the  sign  and  the  thing  signified  is  unmis- 
takable,   has    been  well    set    forth  by    a    living 
author,   greatly   distinguished   for   his   scientific 
attainments.*     Of  the  pure  sciences,  he  says 
Herschei's        "  Their  objects  are  so  definite,  and  our  no- 
"^^*"      tions  of  them  so  distinct,  that  we  can  reason 
about  them  with  an  assurance  that  the  words  and 
signs  of  our  reasonings  are  full  and  true  repre- 
sentatives of  the  things  signified ;    and,  conse- 
Exactian-    qucutly,  that  when  we  use  language  or  signs  in 
rents  error,   argument.  We   neither   by    their  use   introduce 
extraneous  notions,  nor  exclude  any  part  of  the 
case  before  us  from  consideration.     For  exam- 

*  Sir  John  Herschel,  Discourse  on  the  study  of  Natural 
Philosophy. 


CHAP.  I.]  EXACT     TERMS.  353 

pie  :  the  words  space,  square,  circle,  a  hundred,  Mathematical 

tcniis  GXcici 

&c.,  convey  to  the  mind  notions  so  complete 
in  themselves,  and  so  distinct  from  every  thing 
else,  that  we  are  sure  when  we  use  them  we 
know  and  have  in  view  the  whole  of  our  own 
meaning.  It  is  widely  different  with  words  ex-  Different  in 
pressing  natural  objects  and  mixed  relations,  otiier  terms 
Take,  for  instance,  Iron.  Different  persons  at- 
tach very  different  ideas  to  this  word.  One  who 
has  never  heard  of  magnetism  has  a  widely  dif- 
ferent notion  of  iron  from  one  in  the  contrary 
predicament.     The  vulgar  who  regard  this  metal   How  iron  is 

regai-ded  by 

as  incombustible,  and  the  chemist,  who  sees  it  the  chemist 
burn  with   the   utmost  fury,  and  who  has  other 
reasons  for  regarding  it  as  one  of  the  most  com- 
bustible bodies  in    nature ;    the  poet,  who  uses   The  poet 
it  as  an  emblem  of  rigidity  ;  and  the  smith  and 
engineer,  in  whose  hands  it  is  plastic,  and  mould- 
ed like  wax  into  every  form  ;  the  jailer,  who  prizes    xhejaiier: 
it   as   an   obstruction,  and   the    electrician,   who  The  eiectrt 
sees  in  it  only  a  channel  of  open  communication 
by  which  that  most  impassable  of  obstacles,  the 
air,  may  be  traversed  by  his  imprisoned  fluid, — 
have  all  different,  and   all  imperfect  notions   of 
the  same  word.     The  meaning  of  such  a  term   Final  luns 
is  like  the  rainbow — everybody  sees  a  different 
one,  and  all  maintain  it  to  be  the  same." 

"  It  is,  in  factr  lu  this  double  or  incomplete 
23 


354  UTILITY     OF     MATHEMATICS.  [bOOK  III. 

Incomplete   sense  of  words,  that  we  must  look  for  the  origin 

meaning  the       r.  ,  .  ^     ,  .  i  •    . 

souiceof  ^^  ^  very  large  portion  oi  the  errors  into  which 
error.      ^^.^  ^^jj^    Now,  the  study  of  the  abstract  sciences, 

Mathematics  such    as    Arithmetic,    Geometry,    Algebra,   &c., 

such  errors,  whilc  they  aftord  scope  for  the  exercise  of  rea- 
soning about  objects  that  are,  or,  at  least,  may 
be  conceived  to  be,  external  to  us ;  yet,  being 
free  from   these   sources  of  error  and   mistake, 

Requires  a   accustom    US   to  the  strict    use    of   language  as 

strict  use  of  .  ,,„...., 

language,  an  instrument  oi  reason,  and  by  lamuianzing  us 
in  our  progress  towards  truth,  to  walk  uprightly 
and  straightforward,  on  firm  ground,  give  us 
that  proper  and  dignified  carriage  of  mind  which 
Results,  could  never  be  acquired  by  having  always  to 
pick  our  steps  among  obstructions  and  loose 
fragments,  or  to  steady  them  in  the  reeling  tem- 
pests of  conflicting  meanings." 

Two  ways  of         §  373.     Mr.   Lockc  lays    down    two  ways    of  in- 
acquiring 

knowledge,   crcasing  our  knowledge  : 

1st.    "Clear   and   distinct    ideas    with    settled 
names  ;  and, 

2d.  "  The   finding  of  those  which  show  their 
agreement  or  disagreement ;"  that  is,  the  search- 
ing out  of  new  ideas  which  result  from  the  com- 
bination of  those  that  are  known, 
ftrat.  In  regard  to  the  first  of  these  ways,  Mr.  Locke 

says  :  "  The  first  is  to  get  and  settle  in  our  minds 


CHAP.  I.]  INCREASING     KNOWLEDGE.  355 

determined  ideas  of  those   things,  whereof  we     wcasof 
have  general  or  specmc  names  ;  at  least,  oi  so   ^^  ciiatinct 
many  of  them   as   we  would  consider  and  im- 
prove our  knowledge  in,  or  reason  about."  *  *  * 
"  For,  it  being  evident,  that  our  knowledge  can- 
not exceed  our  ideas,  as  far  as  they  are  either  im-     Reason, 
perfect,  confused,  or  obscure,  we  cannot  expect 
to  have  certain,  perfect,  or  clear  knowledge." 

§  374  Now,  the  ideas  which  make  up  our  why  it  is 
knowledge  of  mathematical  science,  fulfil  ex-  matica. 
actly  these  requirements.  They  are  all  im- 
pressed on  the  mind  by  a  fixed,  definite,  and 
certain  language,  and  the  mind  embraces  them 
as  so  many  images  or  pictures,  clear  and  dis- 
tinct in  their  outlines,  with  names  which  sus- 
gest  at  once  their  characteristics  and  properties, 

§  375.    In  the  second  method  of  increasing     second, 
our  knowledge,  pointed  out  by  Mr.  Locke,  math- 
ematical science  offers  the  most  ample  and  the  why  matho 
surest  means.     The  reasonina-s  are  all  based  on  '"^'"=^°^' 

&  the  surest 

self-evident  truths,  and  are  conducted  by  means 
of  the  most  striking  relations  between  the  known 
and  the  unknown.  The  things  reasoned  about, 
and  the  methods  of  reasoning,  are  so  clearly 
apprehended,  that  the  mind  never  hesitates  oi 
doubts.     It  comprehends,  or  it  does  not  compre- 


meaiis. 


356  UTILITY     OF     MATHEMATICS.  [boOK  III. 

hend,  and  the  line  which  separates  the  known 
characteris-  from  the  unknown,  is  always  well  defined.  These 

ticaofthe 

reasoning,    characteristics  give  to  this  system  of  reasoning 
itsadvan-    a  Superiority  over  every  other,  arising,  not  from 
any  difference  in  the  logic,  but  from  a  difference 
in  the  things  to  which  the  logic  is  applied.     Ob- 
servation may  deceive,  experiment  may  fail,  and 
Deinonstra-   experience  prove  treacherous,  but  demonstration 

lion  certain. 

never. 
Mathematics       "  If  it  bc  truc,  then,  that  mathematics  include 

includes  a  ~  ^  .  , 

certain  svs-    ^  pericct   systcm  01   rcasoumg,  whose  premises 
'^"'       are  self-evident,  and  whose  conclusions  are  irre- 
sistible, can  there  be  any  branch  of  science  or 
knowledge   better  adapted   to   the   improvement 
of  the  understanding  ?     It  is   in   this    capacity, 
An  adjunct   as   a  stroug  and  natural  adjunct  and  instrument 
ment  of  rea-  of  rcasou,  that  this  scieuce  becomes  the  fit  sub- 
^"'       ject  of  education  with  all  conditions  of  society, 
whatever  may  be  their  ultimate  pursuits.     Most 
sciences,  as,  indeed,  most  branches  of  knowledge, 
address  themselves  to  some  particular  taste,  or 
subsequent   avocation ;    but  this,  while  it  is  be- 
fore all,  as  a  useful  attainment,  especially  adapts 
itself  to  the  cultivation  and  improvement  of  the 
and  necessa-  thinking  faculty,    and   is   alike  necessary  to  all 
•^  ^  '■*"•     who  would  be  governed  by  reason,  or  live  for 
usefulness."* 

*  Mansfield's  Discourse  on  the  Mathematics. 


CHAP.  I.]  REASONS.  357 


§  376.  The  following,  among  other  consider-    consUera 
ations,  may  serve  to  point  out  and  illustrate  the    ' 7alu°e  of^ 
value   of  mathematical    studies,  as    a    means    of  'n'i''i«'»"tic8, 
mental  improvement  and  development. 

1.    We  readily  conceive    and  clearly    appre-       First. 
hend  the   thmgs  of  which   the   science   treats  ;  ciear  concep. 
they  being  things  simple  in  themselves  and  read-      ^l^^^'^ 
ily  presented  to  the  mind  by  plain  and  familiar 
language.     For  example  :  the  idea  of  number,  of 
one  or  more,  is  among  the  first  ideas  implanted    Example. 
in  the  mind ;  and  the  child  ^vho  counts  his  fin- 
gers or  his  marbles,  understands  the  art  of  num- 
bering them   as  perfectly  as  he   can  know  any 
thing.    So,  likewise,  when  he  learns  the  definition    They  estai> 
01  a  straight  line,  or  a  triangle,  or  a  square,  oi  relations  be- 
a  circle,  or  of  a  parallelosrram,  he  conceives  the  '^^'"^  ''*^^°' 

i  o  '  tions  and 

idea  of  each  perfectly,    and   the  name  and  the      '^'^"'S*. 
image  are  inseparably  connected.     These  ideas, 
so  distinct  and  satisfactory,  are  expressed  in  the 
simplest  and  fewest  terms,  and  may,  if  necessary, 
be  illustrated  by  the  aid  of  sensible  objects. 

2.   The    words    employed    in    the    definitions     Second. 

T  1     •        n  -I  Words  are 

are   always   used    in   the   same    sense — each   ex-  always  used 
pressing   at   all   times   the    same   idea;    so   that  "' ''^''' ^""^^ 

•*■  "^  '  sense. 

when  a   definition   is   apprehended,  the   concep- 
tion of  the  thing,  whose  name  is  defined,  is  per- 
fect ill  the  mind. 
There   is,  therefore,   no   doubt   or    ambiguity 


358  UTILITY     OF     MATHEMATICS.  [bOOK  III 


Hence,  it  is   either  in  the  language,  or  in  regard  to  what  is 

affirmed  or  denied  of  the  things  spoken  of ;  but 

all  is  certainty,  both  in  the  language  employed 

and  in  the  ideas  which  it  expresses. 

Third.  3.    The   sciencB   of  mathematics  employs  no 

no  definition  definition   which    may   not    be    clearly    compre- 

*"" '^'"™  °°' hended — lavs  down  no   axioms    not   universally 

evident  and  •'  •' 

clear.       {yue,  and  to  which  the  mind,   by  the  very  laws 

of  its  nature,  readily  assents ;  and  because,  also, 

in  the  process   of  the  reasoning,  no  principle  or 

truth   is    taken    for  granted,    but  every   link  in 

Theconnec-  the   chain  of  the  argument  is  immediately  con- 

Uon  evident.  i        •   i  i    /^     • ,  •  •  • ,  i 

nected  with  a  definition  or  axiom,  or  with  some 


principle  previously  established. 


Fourth.         4.    The   order   established   in   presenting   the 
atren^tiiens   subjcct    to  the    mind,   aids    the   memory    at  the 

different  fac- 
ulties. 


same  tim.e  that  it  strengthens  and  improves  the 

reasoning   powers.       For   example :    first,    there 

How  ideas    are  the  definitions  of  the  names  of  the  things 

«represen-  ^^j^j^^i^  ^^,q  ^j^g  subjccts  of  the  reasoning;    then 

the  axioms,  or  self-evident  truths,  which,  to- 
gether with  the  definitions,  form  the  basis  of  the 
science.  From  these  the  simplest  propositions 
Hovir  the  de-  ^^6  dcduccd,  and  then  follow  others  of  greater 
nctionsfoi-  (JJfllQ^]^y  •  the  wholc  conuectcd  together  by  rig- 
orous logic — each  part  receiving  strength  and 
light  from  all  the  others.  Whence,  it  follows, 
that  any  proposition  may  be  traced  to  first  prin- 


CHAP.  I.]  SYNTHESIS ANALYSIS.  359 


ciples ;    its    dependence    upon    and    connection  Propositions 

•   I       1  •        •    1  1        1      •  1   •  1         traced  to 

With  those  principles  made  obvious  ;  and  its  truth  their  sources, 
estabhshed  by  certain  and  infalUble  argument. 

5.    The    demonstrative    argument    of  mathe-       ^'*'*- 

Argument 

matics    produces    the    most    certain    knowledge     the  most 

•II  T  certain. 

of  which  the  mind  is  susceptible.  It  estab- 
hshes  truth  so  clearly,  that  none  can  doubt  or 
deny.  For,  if  the  premises  are  certain — that  is.  Reasons, 
such  that  all  minds  admit  their  truth  without 
hesitation  or  doubt,  and  if  the  method  of  draw- 
ing the  conclusions  be  lawful — that  is,  in  accord- 
ance with  the  infallible  rules  of  logic,  the  infer- 
ences must  also  be  true.  Truths  thus  established 
may  be  relied  on  for  their  verity  ;  and  the  knowl-  such  knowi- 

■  J   edge  scienca 

edge    thus    gained    may    well    be    denominated 
Science. 


§  377.  There  are,  as  we  have  seen,  in  mathe-     twosj-s- 
matics,  two  systems  oi  investigation  quite  ditter- 
ent  from  each  other :    the  Synthetical  and  the    synthesis, 
Analytical ;  the   synthetical  beginning  with   the    '^"'^'y*'^- 
definitions   and  axioms,  and   terminating  in   the 
highest  truth  reached  by  Geometry. 

"  This  science  presents  the  very  method  by  synjj,<.ij  .,^, 
which    the    human    mind,  in    its   progress  from 
childhood  to  age,  develops  its  faculties.     What 
first  meets  the  observation   of  a  child  ?     Upon  ^^^^  notiona 
what  are  his  earhest   investigations   employed  ? 


■)60  UTILITY     OF     MATHEMATICS. 


What  is  first  Ncxt  to  coloi',  which  cxists   only  to   the   sight. 

observed.  ... 

figure,  extension,  dimens'on,  are  the  lu'st  objects 
which  he  meets,  and  the  first  which  he  examines. 
He  ascertains  and  acknowledges  their  existence  ; 
then  he  perceives  plurality,  and  begins  to  enu- 
rogressof  merate  ;  finally  he  begins  to  draw  conclusions 
from  the  parts  to  the  whole,  and  makes  a  law 
from  the  individual  to  the  species.  Thus,  he 
has  obtained  figure,  extension,  dimension,  enu- 
meration,  and  generalization.  This  is  the  teach- 
ing of  nature ;  and  hence,  when  this  process 
Process  de-   becomcs  embodied  in  a  perfect  system,  as  it  is 

veloped  in     .        ^^  ,  ,  ,  . 

the  system  of  1'^   Geometry,   that  system   becomes  the   easiest 
Geometry,    ^^^j   most   natural   means    of  strengthening    the 
mind  in  its  early  progress  through  the  fields  of 
knowledge." 
Mrst  neces-        "  Long  after  the  child  has  thus  begun  to  gen- 
ajijUvsIs:    eralize  and  deduce  laws,  he  notices  objects  and 
events,  whose  exterior  relations  afford   no  con- 
clusion  upon   the  subject  of  his  contemplation. 
Machinery  is  in  motion — effects  are   produced, 
lia  method.    He  is   Surprised  ;    examines   and  inquires.      He 
reasons  backward  from  effect  to  cause.     This  is 
Analysis,  the  metaphysics  of  mathematics ;  and 
What  the    through  all  its  varieties — in  Arithmetic — in  Alge- 
scienceis:    ^^^ — ^^^^  -^^  ^j^^  Differential  and  Integral  Calcu- 
lus, it  furnishes  a  grand  armory  of  weapons  for 
acute  philosophical  investigation.     But  analysis 


CHAP.   I.]  BACON    S     OPINION  361 

advances  one  step  further  by  its  peculiar  nota-  whatudoas 
tion;  it  exercises,  in  the  highest  degree,  the  fac- 
ulty of  abstraction,  which,  whether  morally  or 
intellectually  considered,  is  always  connected 
with  the  loftiest  efforts  of  the  mind.  Thus  this 
science  comes  in  to  assist  the  faculties  in  their 
progress  to  the  ultimate  stages  of  reasoning; 
and  the  more  these  analytical  processes  are  cul-      what  it 

finally  ac- 
tivated, the  more   the  mind  looks  in  upon  itself,   compushe^j. 

estimates  justly  and  directs  rightly  those  vast 

powers  which  are  to  buoy  it  up  in  an  eternity 

of  future  being."* 

§  378.  To  the  quotations,  which  have  already 
been  so  ample,  we  will  add  but  two  more. 

"  In    the    mathematics,   I   can  report   no  defi-      Bacon's 

,  ,  ,  i~,        opinion  of 

cience,  except  it  be  that  men  do  not  surh-  matiiematica. 
ciently  understand  the  excellent  use  of  the  pure 
mathematics,  in  that  they  do  remedy  and  cure 
many  defects  in  the  wit  and  faculties  intellectual. 
For,  if  the  wit  be  too  dull,  they  sharpen  it ;  if 
too  wandering,  they  fix  it ;  if  too  inherent  in  the 
sense,  they  abstract  it."f     Again  : 

"  Mathematics  serve  to  inure  and  corroborate     How  the 

I  •      1  Ti-  •  study  of 

the  mind  to  a   constant  diligence  in   study,  to 

*  Mansfield's  Discourses  on  Mathematics 
•j-  Lord  Bacon. 


362  UTILITY     OF     MATHEMATICS.  [bOOK  III. 

mathematics  undcrgo  the  ti'ouble  of  an  attentive  meditation, 

mind.       ^^^  cheerfullj  contend  with  such  difficulties  as 

lie  in   the  way.     They  wholly  deliver  us  from 

credulous    simplicity,    most    strongly   fortify   us 

against  the  vanity  of  skepticism,  effectually  re- 

its  influences.  Strain  US  from  a  rash  presumption,  most  easily 
incline  us  to  due  assent,  perfectly  subjugate  us 
to  the  government  and  weight  of  reason,  and 
inspire  us  with  resolution  to  wrestle  against  the 
injurious  tyranny  of  false  prejudices. 

Howtheyare      "  If  the  fancy  be  unstable  and  fluctuatmg,  it 

6X6  rt  6(1* 

is,  as  it  were,  poised  by  this  ballast,  and  steadied 
by  this  anchor  ;  if  the  wit  be  blunt,  it  is  sharp- 
ened by  this  whetstone ;  if  it  be  luxuriant,  it  is 
pruned  by  this  knife ;  if  it  be  headstrong,  it  is 
restrained  by  this  bridle  ;  and  if  it  be  dull,  it  is 
roused  by  this  spur."* 

§  379.  Mathematics,  in  all  its  branches,  is,  in 
fact,  a  science  of  ideas  alone,  unmixed  with  mat- 
Mathematics  ter  or  material  things ;  and  hence,  is  properly 
apurusci-  ^gj-^^^g^j  g^  Pure  Science.  It  is,  indeed,  a  fairy 
land  of  the  pure  ideal,  through  which  the  mind 
is  conducted  by  conventional  symbols,  as  thought 
is  conveyed  along  wires  constructed  by  the 
hand  of  man. 


*  Dr.  Barrow 


euce. 


CUAP.  I,]  CONCLUSION  363 

§  380.  In  conclusion,  therefore,  we  may  claim    what  mav 
for  the  study  of  Mathematics,  that  it  impresses   claimed  for 
the    mind  with  clear  and   distinct    ideas;  culti- '"''  '^^^^'^ 
vates  habits   of  close   and  accurate    discrimina- 
tion ;    gives,   in   an   eminent  degree,   the  power 
of  abstraction  ;  sharpens  and  strengthens  all  the 
faculties,   and  develops,  to   their  highest  range, 
the  reasoning   powers.       Tlie   tendency  of  this  iia  tendency, 
study  is  to  raise  the  mind  from  the  servile  habit 
of  imitation  to  the  dignity  of  self-reliance   and 
self-action.     It  arms  it  with  the  inherent  ener- 
gies of  its  own  elastic  nature,  and  urges  it  out  Thereaaona. 
on  the  great  ocean   of  thought,   to  niaice  new 
discoveries,  and  enlarge  the  boundaries  ot  men- 
tal etTort. 


364  uTiLiry    of    mathematics.         {book  hi 


CHAPTER    II. 

THE    UTI'i-ITY    OF    MATHEMATICS    REGARDED    AS    A    MEANS    OF    ACQUTRING 
KNOWLEDGE BACONIAN    PHILOSOPHV. 

Mathematics:      §  381.    In  the   preceding  chapter,  we  consid- 
ered the  effects  of  mathematical  studies  on  the 
mind,  merely  as  a  means  of  discipline  and  train- 
How  consid-  ing.     We  regarded  the   study  in  a  single  point 

ered  hereto-  .  .  i  .    mi  r-       i  •  i 

fore:  01  vicw,  VIZ.  as  the  drill-master  oi  the  nitel- 
lectual  faculties  —  the  power  best  adapted  to 
bring  them  all  into  order — to  impart  strength, 
and  to  give  to  them  organization.  In  the 
How  now  present  chapter  we  shall  consider  the  study  un- 
der a  more  enlarged  aspect — as  furnishing  to 
man  the  keys  of  hidden  and  precious  knowl- 
edge, and  as  opening  to  his  mind  the  whole 
volume  of  nature. 


Material  §  382.  The  material  universe,  which  is  spread 

Universe.  . 

out  before  us,  is  the  first  object  of  our  rational 


considered. 


CHAP.  II.]  MATERIAL     UNIVERSE.  365 


regards.    Material  things  are  the  first  with  which 

we  have  to  do.     The  child  plays  with  his  toys   Elements  of 

in    the    nursery,    paddles    in.  the    limpid    water, 

twirls    his    top,    and    strikes   with    the    hammer. 

At  a  maturer  age  a  higher  class   of  ideas  are 

embraced.     The  earth  is  surveyed,  teeming  with 

its   products,    and    filled   with    life.      Man    looks 

around  him  with  wondering  and  delighted  eyes,   obtained  by 

observatioD, 

Ihe  earth  he  stands  upon  appears  to  be  made 
of  firm  soil  and  liquid  waters.  The  land  is 
broken  into  an  irregular  surface  by  abrupt  hills 
and  frowning  mountains.  The  rivers  pursue 
their  courses   through  the   valleys,  without  any    course  of 

nature  5 

apparent  cause,  and  finally  seem  to  lose  them- 
selves in  their  own  expansion.      He  notes   the 
return   of  day  and   night,  at  regular  intervals, 
turns    his  eyes    to  the   starry  heavens,  and  in- 
quires how  far  those  sentinels  of  the  night  may 
be  from  the  world  they  look  down  upon.      He 
is  yet  to  learn  that  all  is  governed  by  general    Governed 
laws  imparted  by  the  fiat  of  Him  who  created     "  laws: 
all  things  ;  that  matter,  in  all  its  forms,  is  sub- 
ject to  those  laws ;  and  that  man  possesses  the    Man  pos- 
capacity  to  investigate,  develop,  and  understand  fg^^^y  ^^  j^ 
them.     It  is  of  the  essence  of  law  that  it  in-  "^''^^^s^^''  ^^ 

understand 

eludes    all   possible    contingencies,    and   insures       ^^'^ 
implicit   obedience;    and  such  are  the  laws  of 
nature. 


366  UTILITY     OF     MATHEMATICS.  j^  BO  OK  III. 


§  383.  To  the  man  of  chance,  nothing  is  more 
mysterious    than    the    developments    of  science. 
Uniformity:  He  does  not  sce  how  so  great   a  uniformity  can 
Variety :     consist  with  the  infinite  variety  which  pervades 
every  department   of  nature.      While    no    two 
individuals  of  a  species    are    exactly  alike,  the 
resemblance   and  conformity  are  so    close,  that 
the  naturalist,  from    the   examination  of  a  sin- 
gle bone,   finds  no  difficulty  in  determining  the 
species,  size,   and  structure  of  the  animal.     So, 
They  appear  also,    in   the  Vegetable    and  mineral   kingdoms: 

in  aU  things. 

all   the  structures  of  growth    or  formation,    al- 
though infinitely  varied,  are  yet  conformable  to 
like  general  laws. 
Science  ne-        This  woudcrful  mcchanism,  displayed  in   the 

cessary  to  /■  •         i  i  •  r        i  i 

thedevei-    structurc  01  animals,  was  but  imperiectly  under- 
"''Tw'  °^    stood,  until  touched  by  the  magic  wand  of  sci- 
ence.    Then,   a  general   law  was  found  to  per- 
vade the  whole.     Every  bone  is  of  that  length 
What  science  and  diameter    best  adapted    to    its    use  ;    every 
muscle  is  inserted  at  the  right  point,  and  works 
about  the  right  centre  ;    the   feathers  of  every 
bird  are  shaped  in  the  right  form,  and  the  curves 
in  which  they  cleave  the   air  are  best   adapted 
What  may   [q  velocity.     It  is  demonstrable,   that  in  every 

be  demon- 
strated,     case,   and   in  all   the  variety  of  forms  in  which 

forces  are   applied,  either  to  increase  power  or 

gain  velocity,  the  very  best  means   have  been 


CHAP.   II.]  PHILOSOPHY     OF     BACON.  367 

adopted  to  produce  the  desired  result.     And  why  why  u  is  so. 
should   it  not    be   so,    since   they   are  enriployed 
by  the  all- wise  Architect  ? 

§  384.  It  is  in  the  investigations  of  the  laws  Applications 
of  nature  that  mathematics  finds  its  widest  Mathematics 
range  and  its  most  striking  applications. 

Experience,  aided  by  observation  and  enlight- 
ened by  experiment,   is  the  recognised  fountain      Bacon's 

Philosophy. 

of  all  knowledge  of  nature.     On  this  foundation 
Bacon  rested  his   Philosophy.     He  saw  that  the 
Deductive    process    of   Aristotle,  in  which   the 
conclusions  do  not   reach  beyond  the  premises,    Aristotie'e: 
was  not  progressive.     It  might,  indeed,  improve 
the    reasoning  pow^ers,   cultivate  habits  of  nice 
discrimination,   and   give    great    proficiency   in 
verbal  dialectics  ;  but  the  basis  was  too  narrow 
for    that    expansive    philosophy,  which    was    to    its  defects. 
unfold   and    harmonize   all   the    laws    of  nature. 
Hence,  he  suggested  a  careful  examination  of  what  Bacon 
nature  in  every  department,  and  laid  the  foun-    ^"^°^*'®  • 
dations  of  a  new  philosophy.     Nature   was    to 
be  interrogated  by  experiment,  observation  was 
to  note  the  results,  and  gather  the  facts  into  the 
storehouse  of  knowledge.      Facts,  so  obtained,  Tiie  moans  tc 
were   subjected    to   analysis    and    collation,    and    ''""^"^'' 
general  laws  inferred  from  such  classification  by 


368  UTILITY     OF     MATHEMATICS.  [bOOK  III. 


Bacon's      a   reasoning  process   called    Induction.     Hence, 
Inductive,    the  system  of  Bacon  is  said  to  be  Inductive. 


New  PhUoso-       §  385.  This  new  philosophy  gave   a  startling 

^  ^*       impulse  to  the    human   mind.     Its   subject  w^as 

Nature — material  and  immaterial ;  its  object,  the 

discovery    and    analysis    of  those   general  laws 

What  it  did.  which  pervade,  regulate,  and  impart  uniformity 
to  all  things  ;  its  processes,  experience,  experi- 
ment, and  observation  for  the  ascertainment  of 
Its  nature,  facts  ;  analysis  and  comparison  for  their  classifi- 
cation ;  and  reasoning,  for  the  establishment  of 

What  aided  general  laws.  But  the  work  would  have  been 
incomplete  without  the  aid  of  deductive  science. 
General  laws  deduced  from  many  separate  cases, 
whatit  by  Induction,  needed  additional  proof;  for,  they 
might  have  been  inferred  from  resemblances  too 
slight,  or  coincidences  too  few.  Mathematical 
science  affords  such  proofs. 

i-he  truths  of      §386.  Regarding  general  laws,  estabhshed  by 
Induction,  as 'fundamental  truths,  expressing  these 
by  means  of  the  analytical  formulas,  and  then 
operating  on  these  formulas  by  the  known  pro- 
How  verified  ccsscs  of  mathematical  science,  we  are  enabled, 

by  Analysis. 

not  only  to  verify  the  truths  of  induction,  but 
often  to  establish  new  truths,  which  were  hidden 
from  experiment    and  observation.     As  ,the  in- 


CHAP.  II.]  EXPERIMENTAL    SCIElfCE.  369 

ductive  process  may  involve  error,  while  the 
deductive  cannot,  there  are  weighty  scientific 
reasons,  for  giving  to  every  science  as  much 
of  the  character  of  a  Deductive  Science  as  pos- 
sible. Every  science,  therefore,  should  be  con-  Asfaras 
structed  with   the  fev^^est  and  simplest  possible 


sciences 


inductions.     These  should  be  made  the  basis  of    *,^V  J 

made  Deduc- 

deductive  processes,  by  w^hich  every  truth,  how-       "^*'- 
ever   complex,    should   be    proved,   even   if  we 
chose  to  verify  the  same  by  induction,  based  on 
specific  experiments. 

§  387.    Every  branch  of  Natural  Philosophy  Natural  Phi- 

■    ■       ^^  ■  ^1  ,  1"  losophy  was 

was  origmally  experimental ;  each  generahza-  expeiimen- 
tion  rested  on  a  special  induction,  and  was  de- 
rived from  its  own  distinct  set  of  observations 
and  experiments.  From  being  sciences  of  pure 
experiment,  as  the  phrase  is,  or,  to  speak  more 
correctly,  sciences  in  which  the  reasonings  con-      is  now 

deductive. 

sist  01  no  more  than  one  step,  and  that  a  step 
of  induction ;  all  these  sciences  have  become, 
to  some  extent,  and  some  of  them  in  nearly  their 
whole  extent,  sciences  of  pure  reasoning :  thus, 
multitudes  of  truths,  already  known  by  induc- 
tion, from  as  many  different  sets  of  experiments, 
have  come  to  be  exhibited  as  deductions,  or  co-  Mathomati- 
rollaries  from  inductive  propositions  of  a  simpler       *^'^*"" 

and  more  universal  character.     Thus,  mechan- 

34 


370  UTILITY     DF     MATHEMATICS.  [bOOK  III 

Deductive    ics,    hydrostatics,    optics,    and    acoustics,    have 

successively  been  rendered   mathematical ;    and 

astronomy  was  brought  by  Newton  within   the 

laws  of  general  mechanics. 

Tiieir  advan-        The   substitution  of  this  circuitous    mode  of 

tages :  .  .  . 

proceeding  for  a  process  apparently  much  easier 

and  more  natural,   is  held,  and  justly  too,  to  be 

the  greatest  triumph  in  the  investigation  of  nature. 

They  rest  on  But,  it  is  iieccssary  to  remark,  that  although,  by 

Inductions.  .  ,,.,..  , 

this  progressive  transiormation,  all  sciences  tend 
to  become  more  and  more  deductive,  they  are 
not,  therefore,  the  less  inductive ;  for,  every  step 
in  the  deduction  rests  upon  an  antecedent  in- 
scicncesde-  ductiou.       The    oppositiou   is,    perhaps,    not    so 

ductive  or  ex- 
perimental,   much  between  the  terms  Deductive  and  Induc- 
tive as  between  Deductive  and  Experimental. 

8  388.   A  science  is  experimental,  in  propor- 

Experimen-  "  i  '  l       i 

tai Science:  ^[q-^-^  ^s  evcry  new  case,  which  presents  any  pe- 
culiar features,  stands  in  need  of  a  new  set  of 
observations    and    experiments,   and  a  fresh  in- 
duction.    It  is  deductive,  in  proportion  as  it  can 
„,^     ,      draw    conclusions,    respecting    cases    of  a   new 

When  de-  '  r  o 

duciive.  l^ind,  by  processes  which  bring  these  cases  un- 
der old  inductions,  or  show  them  to  possess 
known  marks  of  certain  attributes. 

§  389.  We  can  now,  therefore,  perceive,  what 


CHAP.  II.]  DEDUCTIVE     SCIENCES.  371 


is  the  generic  distinction  between  sciences  that    Difference 

etween  Di 
liiclive  and 
lx]:erimenl 
Scienc(^s. 


between  L)ii~ 

can  be  made  deductive  and  those  which   must,   j,, 
as   yet,   remain   experimental.      The   difference  ^gg^",',"'!"''' 
consists   in   our   having  been   able,  or  not  yet 
able,  to   draw  from  first    inductions    as  from  a 
general  law,  a  series  of  connected  and  depend 
ent  truths.      When  this  can    be   done,    the  de 
ductive   process    can    be    applied,  and    the    sci- 
ence becomes  deductive.     For  example :    when   DedncUw 
Newton,  by  observing  and  comparing  the   mo 
tions  of  several  of  the   heavenly  bodies,  discov 
ered   that   all   the   motions,   whether  regular   oi     Example, 
apparently  anomalous,  of  all  the  observed  bodies 
of  the  Solar  System,  conformed  to  the  law  of 
moving   around  a  common   centre,  urged    by  a 
centripetal  force,  varying  directly  as  the   mass, 
and  inversely  as  the  square  of  the  distance  from 
the  centre,  he   inferred  the  existence  of  such  a   whaiN 
law  for  all  the  bodies  of  the  system,  and  then  de- 
monstrated, by  the  aid  of  mathematics,   that  no 
other  law  could  produce  the  motions.     This  is     what  he 
the  greatest  example  which  has  yet  occurred  of     i"'°^^'- 
the   transformation,   at  one  stroke,  of  a  science 
which  was  in  a  great  degree  purely  experimen- 
tal, into  a  deductive  science. 

§  390.  How  far  the  study  of  mathematics  pre-     study  or 
pares   the    mind    for   such   contemplations    and  ""^  """^  "* 


ton  inferred 


372  UTILITY     OF     MATHEMATICS.  [bOOK  III. 


the  such  knowledge,  is  well  set  forth  by  an  old  wri- 


™'"  ■       ter,  himself  a  distinguished  mathematician.     He 


prepares 

ter,  himself  a  disting, 

says  : 

Dr.  Barrow's      «  The  steps  are  guided  by  no  lamp  more  clear- 
opinion. 

ly  through  the  dark  mazes  of  nature,  by  no  thread 

more  surely  through  the  infinite  turnings  of  the 

labyrinth  of  philosophy  ;  nor  lastly,  is  the  bottom 

of  truth  sounded  more  happily  by  any  other  line. 

How       I   will   not   mention   with  how  plentiful  a  stock 

mathematics 

furnish  the  of  knowledge  the  mind  is  furnished  from  these  ; 
with  what  wholesome  food  it  is  nourished,  and 
what  sincere  pleasure  it  enjoys.  But  if  I  speak 
further,  I  shall  neither  be  the  only  person  noi 
the  first,  who  affirms  it,  that  while  the  mind  is 
Abstract  abstracted,  and  elevated  from  sensible  matter, 
it:  distinc-tly  views  pure  forms,  conceives  the  beau- 
ty of  ideas,  and  investigates  the  harmony  of  pro- 
portions, the  manners  themselves  are  sensibly 
corrected  and  improved,  the  affections  composed 
and  rectified,  the  fancy  calmed  and  settled,  and 
the  understanding  raised   and    excited  to  more 

Confirmedby  divine  Contemplations :  all  of  which  I  might  de- 

philosophers.  .  ' 

fend  by  the   authority  and  confirm  by  the  suf- 
frages of  the  greatest  philosophers."* 

Horschers        §  391.    Sir  John  Herschel,  in  his  Introduction 

*  Dr.  Barrow. 


CHAP,   II.]    ASTRONOMY    WITHOUT     MATHEMATICS.     373 

to  his   admirable  Treatise  on   Astronomy,  very     opinions. 

justly  remarks,  that, 

"  Admission  to  its  sanctuary  [the  science  of  Mathemat- 
ical science, 
Astronomy],  and  to  the  privileges  and  feelings    indispensa- 

of  a  votary,  is  only  to  be  gained  by  one  means —  knowiedgeoi 
sound  and  sufficient  knowledge  of  mathematics,  '^*'''°"°™J'- 
the  great  insti^ument  of  all  exact  inquiry,  with- 
out which  no  man  can  ever  make  such  advances 
in  this  or  any  other  of  the  higher  departrnents 
of  science  as  can  entitle  him  to  form  an  inde- 
pendent opinion  on  any  subject  of  discussion 
within  their  range. 

"  It  is   not  without  an  effort   that  those  who     informa- 

tion  caunot 

possess    this    knowledge    can    communicate    on     be  given 
such  subjects  with  those  who  do  not,  and  adapt    °j,avenfr 
their  language  and  their  illustrations  to  the  ne-  matiiematics. 
cessities   of  such   an  intercourse.      Propositions 
which  to  the  one  are  almost  identical,  are  the- 
orems of  import  and  difficulty  to  the  other ;  nor 
is  their  evidence  presented  in  the  same  way  to 
the   mind  of  each.      In   treating   such   proposi-      Except 
tions,  under  such  circumstances,  the  appeal  has  brousmeth- 
to  be  made,  not  to  the  pure  and  abstract  reason,        °  ^' 
but  to   the  sense    of  analogy — to    practice    and 
experience  :  principles  and  modes  of  action  have 
to  be  established,  not  by  direct  argument  from 
acknowledged  axioms,  but  by  continually  refer- 
ring to  the  sources  from  which  the  axioms  them-     Reasons 


374  UTILITY     OF     MATHEMATICS.  [bOOK  III. 

Must  begin   selves  have  been  drawn,  viz.  examples  ;  that  is  tc 

with  the  aim-  ,        ,     .        .  „  ,  i     i         1 1  •  •         . 

piesteie-  ^^7'  "^y  bringing  torvvard  and  dwelling  on  simple 
ments:  ^^^  familiar  instances  in  which  the  same  prin- 
ciples and  the  same  or  similar  modes  of  action 
take  place ;  thus  erecting,  as  it  were,  in  each 
particular  case,  a  separate  induction,  and  con- 
structing at  each  step  a  little  body  of  science  to 

Illustration    meet  its   exigencies.     The  difference  is   that  of 

or  the  differ-      _  _ 

ence  be-  pioneering  a  road  through  an  untraversed  coun- 
Btructionby  ^'^'J'  ^^^'^  advancing  at  ease  along  a  broad  and 
scientific  and  bgatgj^  highway ;  that  is  to  say,  if  we  are  deter- 

unscientific  o  ./    '  j  ' 

methods,     mined  to  make  ourselves  distinctly  understood,, 
and  will  appeal  to  reason  at  all."     Again : 
Mathematics       "  A  Certain  moderate  degree   of  acquaintance' 
necessary  to  ^^j^j^  abstract  scicncc  is  highly  desirable  to  every 

physics:  <o      ^  J 

one  who  would  make  any  considerable  progress 
in  physics.  As  the  universe  exists  in  time  and 
place  ;  and  as  motion,  velocity,  quantity,  num- 
ber, and  order,  are  main  elements  of  our  knowl- 
edge of  external  things  and  their  changes,  an 
acquaintance  with  these,  abstractedly  consid- 
why  it  is  so  ercd  (that  is  to  say,  independent  of  any  consid- 
nec(!ssary.  ^^.j^^^j^jj  ^f  particular  things  moved,  measured, 
counted,  or  arranged),  must  evidently  be  a  use- 
ful preparation  for  the  more  complex  study  of 
nature."* 


*  Sir  John  Herschel  on  the  study  of  Natural  Philosophy, 


CHAP.   II.]  ASTRONOMY.  375 


§  392.  If  we  consider  the  department  of  cheni-  Nccesairy  in 

.    ,  I'll  . ,  •  11  chernistrv. 

istiy, — which  analyzes  matter,  exammes  the  ele- 
ments of  which  it  is  composed,  develops  the  laws 
which  unite  these  elements,  and  also  the  agencies 
which  will  separate  and  reunite  them, — we  shall 
find  that  no  intelligent  and  philosophical  analysis 
can  be  made  without  the  aid  of  mathematics. 

§393,    The  mechanism  of  the   physical   uni-    Laws  long 

uaknowii. 

verse,  and  the  laws  which  govern  and  regulate 
its  motions,  were  long  unknown.  As  late  as  the 
17th  century,  Galileo  was  imprisoned  for  pro-  caiiieo. 
mulgating  the  theory  that  the  earth  revolves  on 
its  axis ;  and  to  escape  the  fury  of  persecution,  h;s  theory. 
renounced  the  deductions  of  science.  Now,  ev- 
ery student  of  a  college,  and  every  ambitious  boy  Now  k 
of  the  academy,  may,  by  the  aid  of  his  Algebra 
and  Geometry,  demonstrate  the  existence  and 
operation   of   those    general   laws   which  enable     ByMimt 

,  .  .  ,  .  ,  ,  ,  means  de- 

hmi   to  trace  with  certainty   the    path  and   mo-   monstrated. 
tions  of  every  body  which  circles  the  heavens. 

§  394.  What  knowledge  is  more  precious,  or  vaiue 
more  elevatina;  to  the  mind,  than  that  which  f  scip»t'fl« 
assures  us  that  the  solar  system,  of  which  the 
sun  is  the  centre,  and  our  earth  one  of  the 
smaller  bodies,  is  governed  by  the  general  law 
of  gravitation ;  that  is,  that  each  body  is  re- 
tained in  its  orbit  by  attracting,  and  being  at- 


cnowD 
to  all; 


376  UTILITY     OF     MATHEMATICS.  [boOK    III. 

What      tracted  by,  all  the  others  ?    This  power  of  attrac- 

it  teaches.        •  i  i  •    i  •      ,^ 

tion,  by  which  matter  operates  on  matter,  is  the 
great  governing  principle  of  the  material  world. 
The  motion  of  each  body  in  the  heavens  de- 
The  things  peiids  on  the  forces  of  attraction  of  all  the 
others  ;  hence,  to  estimate  such  forces — varying 
as  they  do  with  the  quantity  of  matter  in  each 
body,  and  inversely  as  the  squares  of  their  dis- 
tances apart — is  no  easy  problem  ;  yet  analy- 
Auaiysis:  sis  has  solvcd  it,  and  with  such  certainty,  that 
the  exact  spot  in  the  heavens  may  be  marked 
at  which  each  body  will  appear  at  the  expj ration 

What  it  has  of  any  definite  period  of  time.     Indeed,  a  tele- 
done: 

scope   may  be   so   arranged,  that  at   the   end  of 

How  a      that    time  either    one    of    the    heavenly    bodies 

result  might  .,^  ij^iif- 

be  verified   would  prcscnt  itsclf  to  the  field  of  View ;    and 
^raent^'^'     ^^  ^^^  instrument  could  remain  fixed,  though  the 
time  were    a    thousand  years,    the   precise   mo- 
ment would  discover  the  planet  to  the  eye  of  the 
observer,  and  thus  attest  the  certainty  of  science. 

§  395.  But   analysis  has   done  yet   more.     It 

has  not  only  measured   the  attractive  power  of 

Analysis     each   of  the   heavenly  bodies  ;   determined   their 

determines 

iiaiancing     distanccs  from   a  common  point  and  from  each 

forces.  .        ,       ,     .  .„ 

other;  ascertained  their  specific  gravities  and 
traced  their  orbits  through  the  heavens ;  but 
has  also  discovered  the  existence  of  balancing 


CHAP.   II.]      STABILITY     OF     THE     UNIVERSE.  377 

and    conservative    forces,    evincing  the    highest   Evidence  a 
evidence  of  contrivance  and  design.  *^*'^' 

§  396.  A  superficial  view  of  the  architecture  Architecture 
of  the  heavens  might  inspire  a  doubt  of  the  sta-    ens  siiows 
bihty  of  the  entire  system.     The  mutual  action  P''™''"<'"°y 
of  the   bodies   on   each   other  produces  what  is 
called'  an    irregularity   in    their    motions.      The 
earth,  for  example,  in  her  annual  course  around.  Example  of 
the    sun,    is    affected    by  the    attraction    of  the 
moon  and  of  all  the  planets  which  compose  the 
solar  system ;  and  these  attracting  forces  appear 
to   give    an    irregularity  to   her    motions.     The 
moon  in  her  revolutions  around  the  earth  is  also  ofthemoon. 
influenced  by  the  attraction  of  the  sun,  the  earth, 
and  of  all  the  other  planets,  and  yields  to  each  a 
motion  exactly  proportionate   to  the  force  exert- 
ed ;  and  the  same  is  equally  true  of  all  the  bodies  or  the  othe; 

,  .    I  planets. 

which  belong  to  the  system.  It  was  reserved 
for  analysis  to  demonstrate  that  every  supposed 
irregularity  of  motion  is  but  the  consequence  of 
a  general  law  ;  that  every  change  is  constancy, 
and  every  diversity  uniformity.     Thus,   mathe-  Mathematici 

,  .  ,  ,  proves  tho 

maticaJ   science  assures  us  that  our  system  has  permanency 
not  been  abandoned   to  blind  chance,   but    that    '^^^^^y^ 

tern. 

a  superintending  Providence  is  ever  exerted 
through  those  general  laws,  which  are  so  minute 
as  to  govern  the  motions  of  the  feather  as  it  is 


378  UTILITY     OF     mathematics!  [bOOK  III, 


c-.icrauty  of  Wafted  aloDg  on  the  passing  breeze,  and  yet  so 
omnipotent  as  to  preserve  the  stabihty  of  worlds. 


laws. 


§  397.     But   analysis   goes  yet    another    step. 

That  class  of  wandering  bodies,  known  to  us  by 

cometa:     the  name  of  comets,  although  apparently  escaped 

from  their  own  spheres,  and  straying  heedlessly 

What       ihrouffh  illimitable  space,  have  yet  been  pursued 

mathematics  °  i.  j  i 

proves  in  re-  by  the  telcscope  of  the  observer  until   sufficient 

Kard  to  them. 

data  have  been  obtained  to  apply  the  process 
of  analysis.  This  done,  a  few  lines  written  upon 
paper  indicate  the  precise   times   of  their  reap- 

Kesuitsstri-  pearancc.  These  results,  when  first  obtained, 
were  so  striking,  and  apparently  so  far  beyond 
the  reach   of  science  itself,    as   almost  to  need 

Verification,  the  Verification  of  experience.  At  the  appointed 
times,  however,  the  comets  reappear,  and  sci- 
ence is  thus  verified  by  observation. 

Natm-e  §  398.  The  great  temple  of  nature  is  only  to 

cannot  be  in-  i   n  i         i  r  i  •       i         • 

vestigated  be  Opened  by  the  keys  oi  mathematical  science, 
marhemat'ics  ^®  "^^^  pcrhaps  rcach  the  vestibule,  and  gaze 
with  wonder  on  its  gorgeous  exterior  and  its 
exact  proportions,  but  we  cannot  open  the  por-; 
iiiustration.  tal  and  explore  the  apartments  unless  we  use 
the  appointed  means.  Those  means  are  the 
exact  sciences,  which  can  only  be  acquired  by 
discipline  and  severe  mental  labor. 


CHAP.   II.]  RESULTS      OF     SCIENCE.  379 


The   precious  metals  are   not  scattered  pro-     science: 
fusely  over  the  surface  of  the  earth  ;  they  are, 
for  wise  purposes,  buried  in  its  bosom,  and  can 
be  disinterred  only  by  toil   and  labor.     So  with 
science  :  it  comes   not  by  inspiration  ;  it   is  not 
borne  to  us  on  the  wings  of  the  wind ;  it  can    oniy  to  be 
neither  be  extorted  by  power,  nor  purchased  by      smdy: 
wealth  ;  but  is  the  sure  reward  of  diligent  and 
assiduous  labor.     Is  it  worth  that  labor  ?     What    ft  is  worth 

studj. 

is   it   not   worth  ?     It   has  perforated   the  earth, 

and   she   has  yielded  up  her  treasures  ;    it    has      what 

it  has  done 

guided  in  safety  the  bark  of  commerce  over  dis-  &■•  the  wanti 
tant  oceans,  and  brought  to  civilized  man  the 
treasures  and  choicest  products  of  the  remotest 
climes.  It  has  scaled  the  heavens,  and  searched 
out  the  hidden  laws  which  regulate  and  govern 
the  material  universe ;  it  has  travelled  from 
planet  to  planet,  measuring  their  magnitudes,  sur- 
veying their  surfaces,  determining  their  days  and 
nights,  and  the  lengths  of  their  seasons.  It  has 
also    pushed  its  inquiries   into  regions  of  space,       wiiat 

11  1  .  ,       ,  it  has  done 

where  it  was    supposed  that   the   mind   oi    the   to  make  us 
Omnipotent  never  yet  had  energized,  and  there   ''^'i"^^"teci 

i  •'  o  '  with  the  mil 

located   unknown   worlds — calculating  their  di-       ^erse. 
ameters,  and  their  times  of  revolution. 

§  399.    Mathematical   science   is    a   magnetic       How 
telegraph,  which  conducts  the    mind  from    orb  ^'^  «'™'ii«' 


380  UTILITY     OF     MATHEMATICS.  [bOOK  III. 


aid  the      to  orb  througli   the  entire  regions  of  measured 

mind  iu  its 
inquiries : 


space.  It  enables  us  to  weigh,  in  the  balance 
of  universal  gravitation,  the  most  distant  planet 
of  the  heavens,  to  measure  its  diameter,  to  de- 
termine its  times  of  revolution  about  a  common 
centre  and  about  its  own  axis,  and  to  claim  it 
as  a  part  of  our  own  system. 
How  they        In  these  far  reachings  of  the  mind,  the   im- 

enlarge  it:  .         .         ,  ^   ,,  ,,.,., 

agination  has  luU  scope  tor  its  highest  exercise. 
It  is  not  led  astray  by  the  false  ideal  and  fed  by 
illusive  visions,  which  sometimes  tempt  reason 
from  her  throne,  but  is  ever  guided  by  the  de- 
May  be  ductions  of  science;  and  its  ideal  and  the  real 
are  united  by  the  fixed  1-aws  of  eternal  truth. 

Mind  §  400.   There  is  that  within  us  which  d(;lights 

wrfainty!"  ^^  Certainty.  The  mists  of  doubt  obscuxd  the 
mental,  as  the  mists  of  the  morning  do  the  phys- 
ical vision.  We  love  to  look  at  nature  through 
a  medium  perfectly  transparent,  and  to  see  every 
object  in  its  exact  proportions.     The  science  of 

Why       mathematics  is  that  medium  through  which  the 

mathenidtics 

afford  it.  mind  may  view,  and  thence  understand  all  the 
parts  of  the  physical  universe.  It  makes  man- 
ifest all  its  laws,  discovers  its  wonderlul  harmo- 
nies, and  displays  the  wisdom  and  omnipotence 
of  the  Creator. 


CHAP.  III.]  "practical."  381 


CHAPTER    IIL 


THE   UTILltr    OF    MATHEMATICS    CONSIDERED    AS    FURNISHING    THOSE    RULES    OP 
ART    WHICH    MAKE    KNOWLEDGE    PRACTICALLY    EFFECTIVE. 


§  401.  There  is  perhaps  no  word  in  the  Eng-    practical: 
lish  language    less    understood  than   Practical.       Little 

understood. 

By  many  it  is  regarded  as  opposed  to  theoreti- 
cal.    It  has  become  a  pert  question  of  our  day,   its  popular 

signiiicatioa 

"  Whether  such  a  branch  of  knowledge  is  prac- 


tical ?"     "  If  any  practical  good  arises  from  pur-    Questions 

relating  to 

suing  such  a  study?"     "If  it  be  not  full  time  studies  and 
that  old  tomes  be  permitted  to  remain  untouched 
in  the  alcoves  of  the  library,  and  the  minds  of 
the  young  fed  with  the  more  stimulating  food  of 
modern  progress  ?" 


§  402.  Such  inquiries  are  not  to  be  answered    inquiries' 
by  a  taunt.     They  must  be  met  as  grave  ques-    How  to  be 

considered 

tions,  and  considered  and  discussed  with  calm- 
ness.    They  have  possession  of  the  public  mind ;      i^^ir 

influencei 

they  affect  the  foundations  of  education ;    they 


382 


UTILITY     OF     MATHEMATICS.  [bOOK  III. 


Their      influence  aiid  direct  the  first  steps  ;  they  control 
importduco.   ^^^  ^^^^^  elements  from  which  must   spring  the 


systems  of  public  instruction. 


Practical:  §403.  The  term  "practical,"  in  its  common 
Common  acceptation,  that  is,  in  the  sense  in  which  it  is 
often  used,  refers  to  the  acquisition  of  useful 
knowledge  by  a  short  process.  It  implies  a  sub- 
stitution of  natural  sagacity  and  "  mother  wit" 
for  the  results  of  hard  study  and  laborious  effort. 
It  implies  the  use  of  knowledge  before  its  acqui- 
sition ;  the  substitution  of  the  results  of  mere 
experiment  for  the  deductions  of  science,  and 
the  placing  of  empiricism  above  philosophy. 


What  if 
implies 


infhissensB,       §404.  In  ihis  view,  the  practical  is  adverse 
tt  18  oprosed  ^^  sound  learning,  and  directly  opposed  to  real 

to  progress:  ~  ./       i  i 

progress.  If  adopted,  as  a  basis  of  national  edu- 
cation, it  would  shackle  the  mind  with  the  iron 
fetters  of  mere  routine,  and  chain  it  down  to 
the  drudgery  of  unimproving  labor.  Under 
such  a  system,  the  people  would  become  imita- 
tors and  rule-men.  Great  and  original  principles 
would  be  lost  sight  of,  and  the  spirit  of  inves- 
tigation and  inquiry  would  find  no  field  for  their 
legitimate  exercise. 
Kigiit  But   give   to    "  practical"   its    true   and  right 

Signification,  sjgiiification,    and    it   becomes    a    word   of    the 


Conse- 
quences. 


CHAP,   in.]  ILLUSTRATED.  383 


choicest  import.     In  its  right  sense,  it  is  the  best    Best  means 

1-  1  •  I  -111  II  of  applying 

means  oi  makmg  the  true  ideal  the  actual ;  that  i;„owiedge. 
is,  the  best  means  of  carrying  into  the  business 
and  practical  affairs  of  life  the  conceptions  and 
deductions  of  science.  All  that  is  truly  great 
in  the  practical,  is  but  the  actual  of  an  antece- 
dent ideal. 


§  405.  It  is  under  this  view  that  we  now  pro-    Muti.cmati- 

cal  science : 

pose  to  consider  the  practical  advantages  of 
mathematical  science.  In  the  two  preceding 
chapters  we  have  pointed  out  its  value  as  a 
means  of  mental  development,  and  as  affording 
facilities  for  the  acquisition  of  knowledge.  We 
shall  now  show  how  intimately  it  is  blended  itspraciicai 
with  the  every-day  affairs  of  life,  and  point  out 
some  of  the  agencies  which  it  exerts  in  giving 
practical  development  to  the  conceptions  of  the 
mind. 

§  406.  We  begin  with  Arithmetic,  as  this  Arithmetic 
branch  of  mathematics  enters  more  or  less  into  pracucaiiy 
all  the  others.  And  what  shall  we  say  of  its 
practical  utility  ?  It  is  at  once  an  evidence  and 
element  of  c'vilization.  By  its  aid  the  child  in 
the  nursery  numbers  his  toys,  the  housewife 
keeps  her  daily  accounts,  and  the  merchant  sums 
up  his  daily  business.     The  ten  little  characters, 


384  UTILITY     OF     MATHEMATICS.  [bOOK  III. 


which  we  call  figures,  thus  perform  a  very  im- 

Whai  figures  portant  part  in  human  affairs.  They  are  sleepless 
sentinels  watching  over  all  the  transactions  of 
trade  and  commerce,  and  making  known  their 
final  results.     They  superintend  the  entire  busi- 

rheir  value,  uess  aflTairs  of  the  world.  Their  daily  records 
exhibit  the  results  on  the  stock  exchange,  and 
of  enterprises  reaching  over  distant  seas.     The 

Used  by  the  mechanic  and  artisan  express  the  final  results  of 
all  their  calculations  in  figures.     The  dimensions 

In  building,  of  buildiugs,  their  length,  breadth,  and  height,  as 
well  as  the  proportions  of  their  several  parts,  are 
all    expressed    by   figures  before  the   foundation 

Aid  science,  stoucs  are  laid ;  and  indeed,  all  the  results  of 
science  are  reduced  to  figures  before  they  can 
be  made  available  in  practice. 

§  407.   The  rules  and  practice  of  all  the  me- 
chanic arts  are  but  applications  of  mathematical 
Mathematics  scicnce.     The  masou  computes  the  quantity  of 

aseful  in  the 

mechanic  his  materials  by  the  principles  of  Geometry  and 
the  rules  of  Arithmetic.  The  carpenter  frames 
his  building,  and  adjusts  all  its  parts,  each  to 
the  others,  by  the  rules  of  practical  Geometry. 
Fjtampies  The  millwright  computes  the  pressure  of  the 
water,  and  adjusts  the  driving  to  the  driven 
wheel,  by  rules  evolved  from  the  formulas  of 
^  analysis. 


CHAP.   III.]  ILLUSTRATED.  385 

§  408.  Workshops  and  factories  afford  marked   worksiiops 

(.     ,  ...  ,         T  r.  .       1    and  factories 

illustrations  oi  the  utility  and  value  oi  practical    exhibit  ap- 
science.     Here   the  most  difficult  problems  are  plications  oc 

science. 

resolved,  and  the  power  of  mind  over  matter 
exhibited  in  the  most  striking  light.  To  the 
uninstructed  eye  of  a  casual  observer,  confusion 
appears  to  reign  triumphant.     But  all  the  parts     P"its  ad- 

justoxl  on  a 

of  that  complicated  machinery   are  adjusted  to  general  pian. 

each  other,  and  were  indeed  so  arranged,  and 

according   to    a   general    plan,    before    a    single 

wheel   was  formed   by  the   hand  of  the  forger. 

The  power  necessary  to  do  the  entire  work  was      Powei 

first   carefully  calculated,    and   then   distributed       jmd 

throughout  the  ramifications  of  the  machinery. 

Each  part  was  so  arranged  as  to  fulfil  its  office. 

Every  circumference,   and   band,  and  cog,  has 

its   specific   duty    assigned   it.      The   parts   are    Parts  At  in 

.  their  proper 

made  at  different  places,  after  patterns  formed      places. 
by  the  rules  of  science,  and  when  brought  to- 
gether, fit  exactly.     They  are  but  formed  parts 
of  an  entire  whole,  over  which,  at  the  source 
of  power,  an  ingenious  contrivance,  called  the 
Governor,  presides.     His  function  is  to  regulate    Governor' 
the  force  which  shall  drive  the  whole  according 
to  a  uniform  speed.     He  is  so  intelligent,  and 
of  such  delicate  sensibihty,  that  on  the  slightest  I's  functioiui, 
increase  of  velocity,  he  diminishes  the  force,  and 
adds   additional   power  the   moment   the   speed 

^6 


3B6  UTILITY     OF     MATHEMATICS.  [boOK  III. 

All  Is  but  slackens.  All  this  is  the  result  of  mathematical 
science  Calculation.  When  the  curious  shall  visit  these 
exhibitions  of  ingenuity  and  skill,  let  them  not 
suppose  that  they  are  the  results  of  chance  and 
experiment.  They  are  the  embodiments,  by  in- 
telligent labor,  of  the  most  difficult  investigations 
of  mathematical  science. 

§  409.  Another  striking  example  of  the  appli- 
cation of  the  principles  of  science  is  found  in 
steamship:  the  stcamship. 

In  the  first  place,   the  formation  of  her  hull, 

liow  the  hull  so  as   to  dividc  the  waters  with  the  least  resist- 

is  formed.    ^^^.^^  ^j^^  ^^  ^j^g   samc   time  receive  from  them 

the  greatest  pressure  as  they  close  behind  her, 
Ker masts:    IS    uot  an   casy  problcm.      Her    masts    are    all 
,jo^^       to  be  set  at  the  proper  angle,  and  her  sails  so 
adjusted,     ^djustcd  as  to  gain  a  maximum  force.     But  the 
complication    of    her    machinery,     unless    seen 
through  the  medium   of  science,  baffles  investi- 
gation,  and  exhibits    a  startling    miracle.     The 
burning  furnace,  the  immense  boilers,  the  mass- 
Machinery:  ivc   cylinders,   the   huge   levers,  the   pipes,  the 
lifting  and   closing  valves,   and    all  the    nicely- 
adjusted   apparatus,  appear  too   intricate   to    be 
comprehended  by  the  mind  at  a  single  glance. 
The  whole    Yet  in  all  this  complication — in  all  this  variety 

Bonstructed        ~         .       .    ,  ,  ,  ■  .  •  i 

of  principle  and  workmanship,  science  has  ex- 


CHAP,   in.]  ILLUSTRATED.  387 

erted  its  power.     There  is  not  a  cylinder,  whose  according  to 

the  principle!: 

dimensions   were    not    measured  —  not    a  lever,    of  science: 
whose  power  was  not  calculated — nor  a  valve, 
which  does  not  open  and  shut  at  the  appointed 
moment.     There  is  not,  in  all  this  structure,  a      Fioma 

s;eneral  plan. 

bolt,  or  screw,  or  rod,  which  was  not  provided 
for  before  the  great  shaft  was  forged,  and  which 
does  not  bear  to  that  shaft  its  proper  proportion. 
And  when  the  workmanship  is  put  to  the  test,        By 

.  .  what  means 

and  the  power  of  steam  is  urging  the  vessel  on    navigated, 
her  distant  voyage,  science  alone  can  direct  her 
way. 

In  the  captain's  cabin  are  carefully  laid  away, 
for  daily  use,  maps  and  charts  of  the  port  which   ner  charts, 
he  leaves,  of  the  ocean  he  traverses,  and  of  the 
coasts  and  harbors  to  which  he  directs  his  way. 
On  these  are  marked  the  results  of  much  scien-       Their 
tific  labor.     The  shoals,  the  channels,  the  points       ^^.p^^' 
of  danger  and  the  places  of  security,  are  all  in- 
dicated.     Near  by,  hangs   the  barometer,   con-    Barometer: 
structed  from   the   most    abstruse    mathematical 
formulas,  to  indicate   changes   in  the  weight  of 
the   atmosphere,   and    admonish  him  of  the   ap- 
proaching tempest.     On  his  table  lie  the  sextant,     sextant: 
and  the  tables  of  Bowditch.     These  enable  him, 
by  observations  on  the  heavenly  bodies,  to  mark 
his  exact  place  on  the  chart,  and  learn  his  posi-    xiieiruses. 
tion  on  the  surface  of  the  earth.     Thus,  practical 


388 


UTILITY     OF     MATHEMATICS.  [buOK  III. 


Science 

guides  the 

ship  : 


What 
thus  accom- 
plishes. 


Capt.  Hall's 
voyage. 

Its  length: 


and 
incidents. 


Observations 
taken. 


science,  which  shaped  the  keel  of  the  ship  to 
its  proper  form,  and  guided  the  hand  of  the  me- 
chanic in  every  workshop,  is,  mider  Providence, 
the  means  of  conducting  her  in  safety  over  the 
ocean.  It  is,  indeed,  the  cloud  by  day  and  the 
pillar  of  fire  by  night.  Guiding  the  bark  of 
commerce  over  trackless  waters,  it  brings  dis- 
tant lands  into  proximity,  and  into  political  and 
social  relations. 

"  We  have  before  us  an  anecdote  communi- 
cated to  us  by  a  naval  officer,*  distinguished 
for  the  extent  and  variety  of  his  attainments, 
which  shows  how  impressive  such  results  may 
become  in  practice.  He  sailed  from  San  Bias, 
on  the  west  coast  of  Mexico,  and  after  a  voyage 
of  eight  thousand  miles,  occupying  eighty-nine 
days,  arrived  off  Rio  de  Janeiro ;  having  in  this 
interval  passed  through  the  Pacific  Ocean,  round- 
ed Cape  Horn,  and  crossed  the  South  Atlantic, 
without  making  any  land,  or  even  seeing  a  single 
sail,  with  the  exception  of  an  American  whaler 
off  Cape  Horn.  Arrived  within  a  week's  sail 
of  Rio,  he  set  seriously  about  determining,  by 
lunar  observations,  the  precise  line  of  the  ship's 
course,  and  its  situation  in  it,  at  a  determinate 
moment ;    and    having   ascertained    this    within 


*  Captain  Basil  Hall. 


CHAP.   III.]  ILLUSTRATED.  389 


from  five  to  ten  miles,  ran  the  rest  of  the  way  Remarkable 

1        ,1  J  J  1-  .11       coincidence. 

by  those  more  ready  and  compendious  methods, 
known  to  navigators,  which  can  be  safely  em- 
ployed for  short  trips  between  one  known  point 
and  another,  but  which  cannot  be  trusted  in  long      short 

.  methods. 

voyages,  where  the  moon  is  the  only  sure  guide. 

"  The  rest  of  the  tale,  we  are  enabled,  by  his 
kindness,  to  state  in  his  own  words  :  '  We  steered   Particulars 

S^  3.16(1 

towards  Rio  de  Janeiro  for  some  days  after  ta- 
king the  lunars  above  described,  and  having 
arrived  within   fifteen    or   twenty   miles    of  the    Arrival  ai 

Rio. 

coast,  I  hove-to  at  four  in  the  morning,  till  the 
day  should  break,  and  then  bore  up  :  for  although 
it  was  very  hazy,  we  could  see  before  us  a  couple 
of  miles  or  so.  About  eight  o'clock  it  became  so 
foggy,  that  I  did  not  like  to  stand  in  further,  and 
was  just  bringing  the  ship  to  the  wind  again,  be- 
fore sending  the  people  to  breakfast,  when  it  sud- 
denly cleared  off,  and  I  had  the  satisfaction  of  Discovery  of 

Harbor. 

seeing  the  great  Sugar- Loaf  Rock,  which  stands 

on  one  side  of  the  harbor's  mouth,  so  nearly  right 

ahead  that  we  had  not  to  alter  our  course  above 

a  point  in  order  to  hit  the  entrance  of  Rio.     This 

was  the  first  land  we  had  seen  for  three  months,  First  land  in 

after  crossing  so  many  seas,  and  being  set  back-     months, 

wards   and   forwards   by   innumerable   currents 

and  foul  winds.'      The  effect  on  all  on   board      Effect 

might  well  be  conceived  to  have  been  electric  ; 


390  UTILITY     OF     MATHEMATICS.  [boCK   III. 

on  the  crew,  and  it  is  iiecdless  to  remark  how  essentially  the 
authority  of  a  commanding  officer  over  his  crew 
may  be  strengthened  by  the  occurrence  of  such 
incidents,  indicative  of  a  degree  of  knowledge 
and  consequent  power  beyond  their  reach."* 

Surveying.         §  410.    A  useful   application  of  mathematical 
science  is  found  in  the  laying  out  and  measure- 
Measure-     ment  of  land.     The  necessity  of  such  measure- 
ment of  land.  ir-T-i-  I  r  ri  i 

ment,  and  oi  dividing  the  suriace  oi  the  earth 
into  portions,  gave  rise  to  the  science  of  Geom- 
ownership:  ctry.     The  Ownership  of  land  could  not  be  de- 
How       termined  without   some  means  of  running  boun 

determined.  .....  t  n- 

dary  lines,  and  ascertaining  limits.  Levelling 
is  also  connected  with  this  branch  of  practical 
mathematics. 

By  the  aid  of  these  two  branches  of  practical 
science,  we  measure  and  determine  the  area  or 

Contents  of  Contents  of  ground ;  make  maps  of  its  surface  ; 
measure  the  heights  of  hills  and  mountains ; 
Rivers,  find  the  directions  of  rivers ;  measure  their  vol- 
umes, and  ascertain  the  rapidity  of  their  cur- 
rents. So  certain  and  exact  are  the  results,  that 
entire  countries  are  divided  into  tracts  of  con- 
venient  size,  and  the  rights  of  ownership  fully 

Certainty     sccurcd.     The  rulcs  for  mapping,  and  the  con- 

*  Sir  John  Herschel,  on  the  study  of  Natural  Philosophy 


CHAP.  III.]  ILLUSTRATED.  391 


ventional   methods  of  representing    the   surface     Mappinii. 
of  ground,  the  courses  of  rivers,  and  the  heights 
of  mountains,  are  so  well  defined,  that  the  nat- 
ural features  of  a  country  may  be  all  indicated    Features  of 

the  ground. 

on  paper.     Thus,  the  topographical  features  of 

all  the  known  parts  of  the  earth   may  be  cor-  Their  repre- 

rectly  and  vividly  impressed  on  the  mind,  by  a 

map,  drawn  according  to  the  rules  of  art,  by  the 

human  hand. 

§  411.   Our  own   age  has  been  marked  by  a    Runways, 
striking  application  of  science,  in  the  construc- 
tion of  railways.     Let  us  contemplate  for  a  mo-  xhe  problem 
ment  the  elements  of  the  problem  which  is  pre- 
sented in  the  enterprise  of  constructing  a  railroad 
between  two  given  points. 

In  the  fii'st  place,  the  route  must  be   carefully  Examination 

.        .  ,  .       ,  ...  of  their 

examined  to  ascertam  its  general  practicability.  routes. 
The  surveyor,  with  his  instruments,  then  ascer-  surveys. 
tains   all   the  levels   and  grades.     The  engineer 

examines  these  results  to  determine  whether  the  office  of  the 

P  .  ••Ill  engineer. 

power  01  steam,  in  connection  with  the  best 
combination  of  machinery,  will  enable  him  to 
overcome  the  elevations  and  descend  the  decliv- 
ities in  safety.  He  then  calculates  the  curves  calculations 
of  the  road,  the  excavations  and  fillings,  the 
cost  of  the  bridges  and  the  tunnels,  if  there  are 
any ;  and  then  adjusts  the  steam-power  to  meet 


392 


UTILITY     OF     MATHEMATICS.  [bOOK  III. 


Completion  the  coiiditions.  In  a  few  months  after  the  enter- 
prise is  undertaken,  the  locomotive,  with  its  long 
train  of  passenger  and  freight  cars,  rushes  over 
the  tract  with  a  superhuman  power,  and  fulfils 
the  office  of  uniting  distant  places  in  commer- 
cial and  social  relations. 

But  that  which  is  most  striking  in  all  this,  is 
the  fact,  that  before  a  stump  is  grubbed,  or  a 
spade  put  into  the  ground,  the  entire  plan  of  the 
work,  having  been  subjected  to  careful  analysis, 
is  fully  developed  in  all  its  parts.     The  construc- 

The  whole  tiou  is  but  the  actual  of  that  perfect  ideal  which 
the  mind  forms  within  itself,  and  which  can 
spring  only  from  the  far-reaching  and  immuta- 
ble principles  of  abstract  science. 


The  striking 
fact. 


the  result  of 
science. 


§  412.  Among  the  most  useful  applications  of 

practical  science,  in  the  present  century,  is  the 

croton      introduction  of  the  Croton  water  into  the  city 

aqueduct.      ^fj^^^York. 

In  the  Highlands  of  the  Hudson,   about  fifty 

miles  from  the  city,  the  gushing  springs  of  the 

Sources  of    mountains  indicate   the   sources   of   the    Croton 

the  river.         .  i  ■    i 

river,   which    enters    the    Hudson    a    few   miles 

below  Peekskill.     At  a  short  distance  from  the 

Principal     mouth,  a  dam  fifty-five  feet  in  height  is  thrown 

reservoir. 

across   the  river,  creating  an  artificial  lake  for 
the  permanent  supply  of  water.     The  area  of  this 


CHAP.   III.]  ILLUSTRATED.  393 


lake  is  equal  to  about  four  hundred  acres.     The      its  area, 
aqueduct  commences  at  the  C!roton  dam,  on  a   Aqueduct, 
line  forty  feet   above  the  level   of  the  Hudson 
river,  and    runs,  as   near  as   the   nature   of  the 
ground  will  permit,  along  the  east  bank,  till  it 
reaches   its   final    destination   in   the   reservoirs 
of  the  city.     There  are  on  the  line  sixteen  tun-   its  tunnels 
nels,  varying  in   length  from   160  to  1,263  feet, 
making  an  aggregate  length  of  6,841  feet.     The 
heights  of  the  ridges  above  the  grade  level  of  the       Their 

heights. 

tunnels  ran2:e  from  25  to  75  feet.  '  Tvventv-five 
streams  are   crossed  by  the  aqueduct  in  West-      streams 

crossGcl 

Chester  county,  varying  from  ]  2  to  70  feet  below 
the  grade  line,  and  from  25  to  S3  feet  below  the 
top  covering  of  the  aqueduct.  The  Harlem  iiariemrivei 
river  is  passed  at  an  elevation  of  120  feet  above 
the  surface  nf  the  water.  The  average  dimen- 
sions of  the  interior  of  the  aqueduct,  are  about 
seven  feet  in  width  and  eis-ht  feet  in  heio-ht. 

O  O 

The  width  of  the  Harlem  river,  at  the  point    iiswidtiu 
where  the  aqueduct    crosses  it,  is  six  hundred 
and  twenty  feet,  and    the  general  plan  of  the 
bridge  is  as   follows :    There  are  eight  arches,      ^Ms<i  • 
each  of  80  feet  span,  and  seven  smaller  arches, 
each  of  50  feet  span,  the  whole  resting  on  piers 
and  abutments.      The   length    of   the  bridge  is    its  length: 
1,450  feet.     The  height  of  the  river  piers  from 
the  lowest  foundation  is  96   feet.     The   arches 


394  UTILITY     OF     MATHEMATICS.  [bOOK  HI 

Its  height:   are   semi-circulai',   and  the   height  from  the  low- 
est foundation   of   the  piers   to   the  top   of  the 
luwidth.    parapet  is   149  feet.     The  width  across,  on  the 
top,  is  21  feet. 

To  afford  a  constant  supply  of  water  for  dis- 
tribution in  the  city  two  large  reservoirs  have 
Receiving    been  const.ructed,  called  the  receiving  reservoir 

Reservoir: 

and  the  distributing  reservoir.  The  surface  of 
the  receiving  reservoir,  at  the  water-line,  is  equal 

lis  extent,  to  thirtv-onc  acres.  It  is  divided  into  two  parts 
by  a  wall  running  east  and  west.     The  depth  of 

Depth  of    water  in   the   northern  part  is  twenty  feet,  and 

water.         .  ,  .  „ 

in  the  southern  part  thirty  feet. 

Distributing  The  distributing  reservoir  is  located  on  the 
highest  ground  which   adjoins   the   city,   known 

Its  capacity,  as  MuiTay  Hill.     The  capacity  of  this  reservoir 
is  equal  to  20,000,000  of  gallons,  which  is  about 
one-seventh  that  of  the  receiving  reservoir,  and 
the  depth  of  water  is  thirty- six  feet. 
Power  The  full  power  of  science  has  not  yet  been 

illustrated.  A  perfect  plan  of  this  majestic 
structure  was  arranged,  or  should  have  been, 
before  a  stone  was  shaped,  or  a  pickaxe  put  into 
the  ground.  The  complete  conception,  by  a 
single  mind,  of  its  general  plan  and  minutest 
details,  was  necessary  to  its  successful  prosecu- 

Whatitac-  ^jqj-^       j|.   ^^g   -y^^ithin  the  range   and  power  of 

complished.  '  .  °  '■ 

science  to  have  given  the  form  and  dimensions 


CHAP.  III.]  ILLUSTRATED.  395 

of  every  stone,   so  that  each  could  have  been 
shaped  at  the  quarry.     The  parts  are  so  con-     connec- 
nected   by   the   laws  of  the  geometrical    forms,    '""*°  ^"■' 

•'  o  '  parts. 

that  the  dimensions  and  shape  of  each  stone  was 
exactly  determined  by  the  nature  of  that  portion 
of  the  structure  to  which  it  belonged. 

§413.  We  have  presented  this  outline  of  the   view  of  the 
Croton    aqueduct    mainly    for    the    purpose    of    aqueduct: 
illustrating   the    power  and  celebrating   the    tri-   wby  given, 
umphs    of    mathematical  science.      High  intel- 
lect, it  is   true,  can  alone  use  the   means  in  a 
work   so    complicated,   and   embracing  so  great 
a  variety  of  intricate  details.     But  genius,  even     uttieac- 
of  the  highest  order,  could  not  accomplish,  with-   '=°'"p''^^^'1 

o  '  r         '  without 

out   continued    trial    and    laborious    experiment,      science. 
such   an   undertaking,  unless   strengthened    and 
guided  by  the  immutable  truths  of  mathematical 
science. 

§  414.  The  examination  of  this  work  cannot  what 
but  fill  the  mind  with  a  proud  consciousness  of  ^5",,^ 
the  power  and  skill  of  man.  The  struggling 
brooks  of  the  mountains  are  collected  together — 
accumulated — conducted  for  forty  miles  through 
a  subterranean  channel,  to  form  small  lakes  in 
the  vicinity  of  a  populous  city. 

Fi'om  these  sources,  by  an  unseen  process,  the 


396  UTILITY     OF     MATHEMATICS.  [bOOK  lit. 


pure  water  is  carried  to  every  dwelling  in  the 

large  metropolis.      The  turning  of  a  faucet  de- 

cons(v      livers  it  from  a  spring  at  the  distance  of  fifty 

quencea 

which  have  milcs,  as  purc  as  when  it  gushes  from  its  granite 
hills.  That  unseen  power  of  pressure,  which 
resides  in  the  fluid  as  an  organic  law,  exerts  its 
force  with  unceasing  and  untiring  energy.  To 
minds  enlightened  by  science,  and  skill  directed 
by  its  rules,  we  are  indebted  for  one  of  the  no- 
blest works  of  the  present  century.      May   we 

conciucion.  not,  therefore,  conclude  that  science  is  the  only 
sure  means  of  giving  practical  development  to 
those  great  conceptions  wliich  confer  lasting 
benefits  on  mankind?  "All  that  is  truly  great 
in  the  practical,  is  but  the  result  of  an  antece- 
dent ideal." 


INDEX. 


Absteaction  . . .  That  faculty  of  the  mind  which  enables  us,  in  con. 
templating  any  object  to  attend  exclusively  to  some 
particular  circumstance,  and  quite  withhold  our  at- 
tention from  the  rest,  Section  12. 
"  Is  used  in  three  senses,  13. 

Abstract  Quantity,  107. 

Addition, Spelling  and  reading  in,  122-127. 

"  Examples  in,  159. 

Definitions  of,  207 
"  One  principle  governs  all  operations  in,  236. 

^tna, How  far  designated  by  the  term  mountain,  20. 

A  Geometrical      Proportion,  176. 

AliGEBRA, A  species  of  Universal  Arithmetic,  in  which  letters 

and  signs  are  employed  to  abridge  and  generalize 
all  processes  involving  numbers,  284. 
"  Divided  into  two  parts,  284. 

"  Difficulties  of,  from  what  arising,  290. 

"  Principles  of,  deduced  from  definitions  and  axioms,  301. 

"  Should  pi-ecede  Geometry  in  instruction,  358. 

Alphabet  of  the  language  of  numbers,  91,  120,  121. 

"  Language  of  Arithmetic,  formed  from,  196. 

Analytical  Form,  for  what  best  suited,  71,  95. 

Analysis A  term  embracing  all  the  operations  that  can  be  per- 
formed on  quantities  represented  by  letters,  93,  94, 
95,  278,  377. 
"  It  also  denotes  the  process  of  separating  a  complex 

whole  into  its  parts,  95. 
«  of  problems  in  Arithmetic,  183, 184. 

«■♦  Three  branches  of,  283,  289,  290. 

"  First  notions  of,  how  acquired,  367. 

"  Problems  it  has  solved,  396,  397. 

Angles Eight  angle,  the  unit  of,  254. 

"  A  class  of  Geometrical  Magnitudes,  85,  277. 


398 


IISTDES. 


Apotliecaries'        Weiglit — Its  units  and  scale,  Section  145. 
Apprehension  .  Simple  apprehension  is  tlie  notion  (or  conception)  of 
an  object  in  tlie  mind,  7. 
"  Incomples  apprehension  is  of  one  object  or  of  sev- 

eral without  any  relation  being  perceived  between 
them,  7. 
"  Complex  is  of  several  with  such  a  relation,  7. 

Aeea  or Contents,  Number  of  times  a  surface  contains  its 

unit  of  measure,  148. 
Argument  with   one   premise   suppressed  is  called  an   Enthy- 

meme,  47. 
"  Two  kinds  of  objections,  47. 

"  Every  valid,  may  be  reduced  to  a  syllogism,  52. 

"  at  full  length,  a  syllogism,  56. 

"  concerned  with  connection  between  premises  and  con* 

elusion,  57. 
"  Where  the  fault  (if  any)  lies,  69. 

Arguments,  In  reasoning  we  make  use  of,  42. 

"  Examples  of  unsound,  50. 

"  Rules  for  examining,  70. 

Aristotle  did  not  mean  that  arguments  should  always  be  stated 

syllogistically,  53. 
"  accused  of  darkening  his  demonstrations  by  the  use 

of  symbols,  57. 
"  His  philosophy  not  progressive,  384. 

Aeistotle's  Dictum — Whatever  is  predicated  (that  is,  affirmed  or 
denied)  iiniversally ,  of  any  class  of  things, 
may  be  predicated,  in  like  manner  (viz. 
affirmed  or  denied),  of  any  thing  com- 
prehended in  that  class,  54. 
"  "         Keystone  of  his  logical  system,  54. 

*•  "         Objections  to,  54, 55. 

"  "         a  generalized  statement  of  all  demonstra- 

tion, 55. 
"  "         applied  to  terms  represented  by  letters,  56. 

"  "         not  complied  with,  59,  60. 

"  "         All  sound  arguments  can  be  reduced  to  the 

form  to  which  it  applies,  65,  63. 
Arithmetic  ...  .Is  both  a  science  and  an  art,  180. 

"  It  is  a  science  in  all  that  relates  to  the  properties,  laws, 

and  proportions  of  numbers,  180. 


IKDEX. 


399 


AniTHMETic  ...  .It  is  an  art  in  all  that  concerns  tlieir  application,  Sec- 
tion 181. 
"  Processes  of,  not  affected  by  the  nature  of  the  ob- 

jects, 43. 
"  Illustration  from,  45. 

"  How  its  principles  should  be  explained,  182. 

"  Its  requisitions  as  an  art,  185. 

*  Faculties  cultivated  by  it,  188. 

"  Combinations  in,  196-303. 

"  "What  its  study  should  accomplish,  210. 

"  Art  of,  its  importance,  210. 

"  Elementary  ideas  of,  learned  by  sensible  objects,  211. 

"  Principles  of,  how  they  should  be  taught,  212. 

"  First,  wliat  it  should  accomplish,  218. 

"  "      arrangement  of  lessons,  218-227. 

"  "      what  should  be  taught  in  it,  230. 

"  Second,  should  be  complete  and  practical,  231. 

"  "  arrangement  of  subjects,  232. 

"  "  introduction  of  subjects,  233. 

"  "  reading  of  figures  should  be  constantly  prac- 

tised, 234. 
"  Third,  the  subject  now  taught  as  a  science,  285. 

"  "        I'equirements  from  the  pupil  for,  235. 

"  "        Reduction  and  the  ground  rules  brought  un- 

der one  principle,  236. 
"  "        design  of, — methods  must  differ  from  smaller 

works,  237. 
"  "        examples  in  the  ground  rules,  238. 

"  "        what  subjects  should  be  transferred  from  ele- 

mentary works,  239. 
"  Practical  utility  of,  406,  407. 

"  Should  not  be  finished  before  Algebra  is  commenced, 

356. 
Arithmetical         Proportion,  171. 
Ratio,  171. 

AnT .The  application  of  knowledge  to  practice,  22. 

"  Its  relations  to  science,  22. 

"  A  sing]e  one  often  formed  from  several  sciences,  22. 

of  Arithmetic,  181,  185,  190. 
Astronomy  brought  by  Newton  within  the  laws  of  mechanics,  387. 

"  How  it  became  deductive,  389. 


400 


INDEX. 


Astronomy, 
Authors, 


Auxiliary 
Avoirdupois 

Axiom 

Axioms 


Mathematics  necessary  in,  Section  391. 
methods  of  finding  ratio,  173,  178. 

"         of  placing  Rule  of  Three,  195. 
quotations  from,  on  Arithmetic,  205,  208. 
definition  of  proportion,  272. 
Quantities,  263,  264. 
Weight,  its  units  and  scale,  143. 
.A  self-evident  truth,  27,  100. 
of  Geometry,  process  of  learning  them,  27. 
or  canons,  for  testing  the  validity  of  syllogisms,  67. 
of  Geometry  established  by  Induction,  73. 
for  forming  numbers,  78. 
for   comparison  relate   to   eqtiality  and    inequality, 

109. 
for  inferring  equality,  109,  262,  264,  268. 
'{  "        inequality,  109. 

employed  in  solving  equations,  282,  315. 


Bacon,  Lord, 


Barometer, 
Barrow,  Dr., 
Belief 


Blakewell, 
Bowditch, 
Bkeadth . , 
Bridge, 


Quotation  from,  378. 

Foundation  of  his  Philosophy,  384;  its  subject  Na- 
ture, 385. 
His  system  inductive,  384. 
Object  and  means  of  his  philosophy,  385. 
Construction  and  use  of,  409. 
Quotation  from,  378,  390. 
essential  to  knowledge,  23. 
and  disbelief  are  expressed  in  propositions,  36. 
steps  of  his  discovery,  32. 
Tables  of,  used  in  Navigation,  409. 
.A  dimension  of  space,  81. 
Harlem,  description  of,  412. 


Calculus, In  its  general  sense,  means  any  operation  performed 

on  algebraic  quantities,  285,  286. 
Differential  and  Integral,  287,  288,  289. 
Canons  for  testing  the  validity  of  syllogisms,  67. 

Cause  and  effect,  their  relation  the  scientific  basis  of  induc- 

tion, 33. 
Chemist,  Illustration,  53  ;  idea  of  iron,  372. 

Chemistry  aided  by  Mathematics,  392. 

Circle . .  .A  portion  of  a  plane  included  within  a  curve,  all  the 


INDEX. 


401 


points  of  which  are  equally  distant  from  a  certain 
point  within  called  the  centre.  Section  248.  , 

Circle The  only  curve  of  Elementary  Geometry,  248. 

Property  of,  260. 
Circular  Measure,  its  units  and  scale,  156. 

Classes Divisions  of  species  or  subspecies,  in  which  the  char- 
acteristic is  less  extensive,  hut  more  full  and  com- 
plete, 16. 
The  arrangement  of  objects  into  classes,  with  refer- 
ence to  some  common  and  distinguishing  charac- 
teristic, 16. 
Basis  of,  may  be  chosen  arbitrarily,  20. 
of  a  letter,  295  ;  of  a  product,  296. 
Differential,  287,  288. 

should  be  exhibited  to  give  ideas  of  numbers,  140. 
in  Arithmetic,  196-203. 
taught  in  First  Arithmetic,  220-222. 
Problem  with  reference  to,  397. 
Knowledge  gained  by,  106. 
Reasoning  carried  on  by,  25,  311. 
The  third  proposition  of  a  syllogism,  40. 
in  Induction,  broader  than  the  premises,  31. 
deduced  from  the  premises,  40,  41,  46,  47,  49. 
contradicts  a  known  truth,  in  negative  demonstra- 
tions, 268,  269. 
Quantity,  107. 
causal,  illative,  48. 

denote  cause  and  effect,  premise  and  conclusion,  48. 
Consecutive  values,  326. 

Quantities  which  preserve  a  fixed  value  throughout 
the  same  discussion  or  investigation,  286,  287,  317. 
"  represented  by  the  first  letters  of  the  alphabet,  288. 

Continuity Continuity  defined,  321,  325. 

"  Consequeiices  of  the  law  of  continuity,  328. 

Copula That  part  of  a  proposition  which  indicates  the  act  of 

judgment,  38. 
"  must  be  "  is"  or  "  is  not,"  38,  39. 

Cousin,  quotation  from,  168. 

Curves,  circumference  of  circle  the  simplest  of,  243. 

Croton  river,  its  sources,  412. 

"  dam,  its  construction,  412  ;  lake,  area  of,  413. 


Classification 


Coefficient 

Coins 
Combinations 

Comets, 
Comparison, 

Conclusion  . 


Concrete 
Conjunctions 

Consecutive 
Constants  . . 


402 


IKDEX. 


Croton 


aqueduct,  description  of,  Section  412. 


Decimals,  language  of,  and  scale  for,  164,  165. 

Deduction A  process  of  reasoning  by  wliicli  a  particular  truth 

is  inferred  from  otlier  truths  wliicli  are  known  or 
admitted,  34. 
"  Its  formula  tlie  syllogism,  34. 

Deductive  Sciences,  wliy  they  exist,  101. 

"  "        aid  they  give  in  Induction,  385. 

Definition A  metaphorical  word,  which  literally  signifies  laying 

down  a  boundary,  1. 
"  Is  of  two  kinds,  1. 

"  Its  various  attributes,  2-5. 

Definitions,  General  method  of  framing,  3. 

"  Rules  for  framing,  5  (Note). 

"  and  axioms,  tests  of  truth,  100,  102. 

"  signs  of  elementary  ideas,  204. 

"  Necessity  of  exact,  204. 

Demonstration.  A  series  of  logical  arguments  brought  to  a  conclusion, 
in  which  the  major  premises  are  definitions,  axioms, 
or  propositions  already  established,  241, 
of  a  demonstration,  55. 
to  what  applicable,  242. 
of  Proposition  I.  of  Legendre,  262. 
positive  and  negative,  266-269. 
produces  the  most  certain  knowledge,  376. 
Descartes,  originator  of  Analytical  Geometry,  285. 

Dictum,  Aristotle's,  54,  55,  G6. 

Differential  and  Integral  Calculus.  The  science  which  notes 
the  changes  that  take  place  according  to  fixed 
laws  established  by  algebraic  formulas,  when 
those  changes  are  indicated  by  certain  marks 
drawn  from  the  variable  symbols,  287.  Chap.  V., 
Art.  320. 
**  Coefficients — Marks  drawn  from   the  variable   sym- 

bols, 287,  288. 
"  and  Integral   Calculus — Difference    between   it  and 

Analytical  Geometry,  288. 
"     ■  "  "  "  What  persons  should  study 

it,  387. 
Discussion  of  an  Equation,  313. 


IKDEX. 


403 


Distribution.  .  .A  term  is  distributed,  wlien  it  stands  for  all  its  signi- 
cates.  Section  61. 
"  A  term  is  not  distributed  when  it  stands  for  only  a 

part  of  its  significates,  61. 
Distribution,         Words  wliicli  mark,  not  always  expressed,  62. 
Division,  Readings  in,  130  ;  examples  in,  162. 

"  Combinations  in,  200. 

"  All  operations  in,  governed  by  one  principle,  236. 

"  of  quantities,  how  indicated,  298. 

Dry  Measure,        Its  units  and  scale,  154. 
Duodecimal  Units,  149-151. 


Equality 


Equation 


English  Money,    Its  units  and  scale,  142. 

Enthymeme  . . .  .An  argument  with  one  premise  suppressed,  47. 

Equal Two  geometrical  figures  are  said  to  be  equal  when 

they  contain  the  same  unit  an  equal  number  of 
times ;  and  equal  in  aU  their  parts,  when  they  can 
be  so  applied  to  each  other  as  to  coincide  through- 
out their  whole  extent,  259. 
.Expresses  the  relation  between  two  quantities,  when 
each  contains  the  same  unit  an  equal  number  of 
times,  259,  316. 
.An  analytical  formula  for  expressing  equality,  311- 
316. 
"  A  proposition  expressed  algebraically,  in  which  equal- 

ity is  predicated  of  one  quantity  as  compared  with 
.  another,  313. 
"  either  abstract  or  concrete,  314. 

Equations,-  Subject  of,  divided  into  two  parts,  312. 

"  Five  axioms  for  solving,  315. 

Examples  In  ground  rules  of  Third  Arithmetic,  238. 

"  Of  little  use  to  vary  forms  of,  without  changing  the 

principles  of  construction,  240. 
Experiment,  In  what  sense  used,  25  (Note). 

Exponent An  expression  to  show  how  many  equal  factors  arc 

employed,  297. 
Extremes.  Subject  and  predicate  of  a  proposition,  38,  67. 


Fact. 


.  Any  thing  which  has  been  or  is,  24. 
Knowledge  of,  how  derived,  25. 
In  what  sense  used,  25. 


404  IKDEX, 


Fact regarded  as  a  genus.  Section  25. 

Factories,  Value  of  science  in,  408. 

Fallacy  ...... .Any  unsound  mode  of  arguing  wMcli  appears  to  de- 
mand our  conviction,  and  to  be  decisive  of  the  ques- 
tion in  hand,  when  in  fairness  it  is  not,  68. 

"  Illustration  of,  53. 

"  Example  and  analysis  of,  59,  60. 

"  Material  and  Logical,  69. 

Rules  for  detecting,  70. 
FiGTTRE A  portion  of  space  limited  by  boundaries,  83. 

"  Each  geometrical,  stands  for  a  class,  281. 

Figures In  Arithmetic  show  how  many  times  a  unit  is  taken, 

132. 

"  do  not  indicate  the  kind  of  unit,  132. 

"  Laws  of  the  places  of,  133,  134. 

"  have  no  value,  135,  205. 

"  Methods  of  reading,  137  ;  of  writing,  203. 

"  Definitions  of,  205,  206. 

"  should  be  early  used  in  Arithmetic,  223. 

First  Arithmetic,  what  should  be  taught  in  it,  230. 

"  Faculties  to  be  cultivated  by  it,  218. 

"  Construction  of  the  lessons,  218-222. 

"  Lesson  in  Fractions,  224-228. 

"  Tables  of  Denominate  Numbers — Examples,  229. 

Fractions  Come  from  the  unit  one,  139. 

"  should  be  constantly  compared  with  one,  170. 

Definitions  of,  208. 

"  Lessons  in,  in  First  Arithmetic,  224-228 

Fractional         units,  163 ;  orders  of,  164;  language  of,  164-167,  201. 

"  "      three  things  necessary  to  their  apprehension, 

168. 

"  "      advantages  of,  169. 

"  "      two  things  necessary  to  their  being  equal,  169. 

Galileo,  Imprisoned  in  the  17th  century,  393. 

GENEBALizATiON...The  process  of    contemplating   the    agreement   of 
several  objects  in  certain  points,  and  giving  to  all 
and  each  of   these   objects  a  naine   applicable  to 
them  in  respect  to  this  agreement,  14. 
"  implies  abstraction,  14. 

Genus The  most  extensive  term  of  classification,  and  conse- 


INDEX. 


405 


Genus, 


Geometrical 


Geometry 


Governor, 
Grammar 
Gravitation, 


quently  the  one  involving  the  fewest  particulars. 
Sections  16,  17. 
.Highest.     That  which  cannot  be  referred  to  a  more 
extended  classification,  19. 
Subaltern.     A  species  of  a  more  extended  classifi- 
cation, 18. 
Magnitudes,  three  classes  of,  242,  277. 
"  do  not  involve  matter,  251, 

"  their  boundaries  or  limits,  251. 

"  each  has  its  unit  of  measure,  256. 

"  analysis  of  comparison,  274,  275. 

"  to  what  the  examination  of  properties 

has  reference,  277. 
Proportion,  171 ;  Ratio,  171 ;  Progression,  178. 
.Treats  of  space,  and  compares  portions  of  space  with 
each  other,  for  the  purpose  of  pointing  out  their 
properties  and  mutual  relations,  241. 
Why  a  deductive  science,  261. 
First  notions  of,  how  acquired,  368-370. 
Practical  utility  of,  407. 
Origin  of  the  science,  410. 
Its  place  in  a  course  of  instruction,  359. 
Analytical,  Examines  the  properties,  measures,  and 
relations  of  the  Geometrical  Magni- 
tudes  by   means   of    the    analytical 
symbols,  285,  286. 
"  originated  with  Descartes,  285. 

"  difference  between  it  and  Calculus,  288. 

"  its   importance,   extent,  and    methods, 

^  386. 
Descriptive.  That  branch  of  mathematics  which 
considers  the  positions  of  the  Geo- 
metrical Magnitudes  as  they  may 
exist  in  space,  and  determines  these 
positions  by  referring  the  magni- 
tudes to  two  planes  called  the 
Planes  of  Projection,  862. 
"  how  regarded  in  France,  362. 

Functions  of,  in  machinery,  408. 
Defined,  120. 
Law  of ,  32,  394. 


406 


IKDEX. 


Hall,  Captain's,     voyage  from  San  Bias  to  Eio  Janeiro,  Section  404. 
Harlem  river,        Bridge  over,  and  -width,  412. 
Herscliel,  Sir  John,  Quotation  from,  27,  372,  395, 409. 
Hull  of  the  steamship,  how  formed,  409. 

Illative  Conjunctions,  48. 

Illicit  Process. .When  a  term  is  distributed  in  the  conclusion  which 

Avas  not  distributed  in  one  of  the  premises,  67. 
Indefinite  Propositions,  62. 

Index  Of  a  root,  299. 

Induction  Is  that  part  of  Logic  which  infers  truths  from  facts, 

30-33. 
"  Logic  of,  30. 

"  supposes  necessary  observations  accurately  made,  32. 

"  Example  of,  Blakewell,  32 ;  of  Newton,  32. 

"  based  upon  the  relation  of  cause  and  efEect,  33. 

"  Reasoning  from  particulars  to  generals,  34. 

"  its  place  in  Logic,  72. 

"  how  thrown  into  the  form  of  a  syllogism,  74,  102. 

Truths  of,  verified  by  Deduction,  385,  386. 
Inertia  proportioned  to  weight,  272. 

Infinity,. The  limit  of  an  increasing  quantity,  306-310. 

Integral  Numbers,  why  easier  than  fractions,  170. 

"  constructed  on  a  single  principle,  235. 

Intuition Is  strictly  applicable  only  to  that  mode  of  contempla- 
tion, in  which  we  look  at  facts,  or  classes  of  facts, 
and  immediately  apprehend  their  relations,  27. 
Iron,  different  ideas  attached  to  the  word,  372. 

Judgment  Is  the  comparing  together  in  the  mind  two  of  the 

notions  (or  ideas)  which  are  the  objects  of  appre- 
hension, and  pronouncing  that  they  agree  or  dis- 
agree, 8. 
"  is  either  Affirmative  or  Negative  8. 


Kant,  Quotation  from,  21. 

Knowledge         Is  a  clear  and  certain  conception  of  that  whicli  is 
true,  23. 
"  facts  and  truths  elements  of,  25. 

of  facts,  how  derived,  25. 
"  some  possessed  antecedently  to  reasoning,  29. 


IISTDEX. 


407 


Knowledge,       tlie  greater  part  matter  of  inference,  Section  29. 
"  two  ways  of  increasing,  373. 

"  cannot  exceed  our  ideas,  373. 

"  tlie    increase    of,    renders     classification    necessary, 

page  20. 

Language  Affords  the  signs   by  wiiicli  tlie  operations  of    the 

mind  are  recorded,  expressed,  and  communicated, 
10. 
"  Every  branch  of  knowledge  has  its  own,  11. 

*'  of  numbers,  91 ;  of  mathematics,  90. 

"  of  mathematics  must  be  thoroughly  learned,  88. 

"  "  "  its  generality,  89. 

for  fractional  units,  164,  167,  207. 
Arithmetical,  196-203. 
"  exact,  necessary  to  accurate  thought,  209. 

"  of  Arithmetic,  its  uses,  223. 

"  of  Algebra,  the  first  thing  to  which  the  pupil's  mind 

should  be  directed,  294. 
"  Culture  of  the  mind  by  the  use  of  exact,  372. 

"  of  Calculus,  346. 

Laws  of  Nature,  Science  makes  them  known,  21,  319. 

"  "        refers  individual  cases  to  them,  55. 

"  generalized  facts,  55. 

"  include  all  contingencies,  372. 

"  every  diversity  the  effect  of,  396. 

Lemma,  Newton's,  329-337. 

Length  one  dimension  of  space,  80. 

Lessons  in  First  Arithmetic,  how  arranged,  218. 

"  "      "  "  their  connections,  222. 

Letter  may  stand  for  all  numbers,  280. 

"  represents  things  in  general,  281. 

Levelling The  application  of  the  principles  of  Trigonometry  to 

the    determination  of  the  difference   between  the 
distances  of  any  two  points  from  the  centre  of  the 
earth,  361. 
"  Its  practical  uses,  351. 

Limit,  Definition  of,  322. 

of  discontinuous  quantity,  323. 
"  of  continuous  quantity,  325.  ^ 

«  differently  defined,  339-343. 


408  IITDEX. 


Line One  dimension  of  space.  Sections  82,  243. 

"  A  straiglit  line    does   not   change   its   direction,  82, 

243,  368. 
"  Curved  line,  one  wliicli  changes  its  direction  at  every 

point,  82,  243. 
"  Axiom  of  the  straight,  243. 

Lines,  limits  of,  251. 

"  Auxiliary,  263. 

Liquid  Measure,  Its  units  and  scale,  153. 
"  Local  value  of  a  figure,"  has  no  significance,  135,  205. 
Locke,  Quotation  from,  373. 

Logic Takes  note  of  and  decides  upon  the  sufficiency  of  the 

evidence  by  which  truths  are  established,  29. 
"  Nearly  the  whole  of  science  and  conduct  amenable 

to,  29. 
"  of  Induction,  its  nature,  30. 

"  Archbishop  Whately's  views  of,  72. 

"  Mr.  Mill's  views  of,  72. 

Logical  Fallacy,   69. 

Machinery  of  factories  arranged  on  a  general  plan,  408. 

'■'  of  the  steamship,  409. 

Major  Premise,     often  supj^ressed,  cannot  be  denied,  46. 
"  ultimate,  of  Induction,  74,  102. 

Major  Premises,  of  Geometry,  241,  261. 

Mansfield,  Mr.,      Quotation  from,  375,  377. 

Mark The  evidence  contained  in  the  attributes  implied  in 

a  general  name,  by  which  we  infer  that  any  thing 
called  by  that  name  possesses  another  attribute  or 
set  of  attributes.  For  example  :  "  All  equilateral 
triangles  are  equiangular."  Knowing  this  general 
proposition,  wlie^  we  consider  any  object  possess- 
ing the  attributes  implied  in  the  term  "equila- 
teral triangle,"  we  may  infer  that  it  possesses  the 
attributes  implied  in  the  term  "  equiangular ; " 
thus  using  the  first  attributes  as  a  mark  or  evi- 
dence of  the  second.  Hence,  whatever  jaossesses 
any  mark  possesses  those  attributes  of  which  it  is 
a  mark,  101,231,263. 

Masts,  of  the  steamship,  how  placed,  409. 

Material  Fallacy,  69. 


INDEX". 


409 


Mathematical 


Mathematics  . 


Measure 


Middle  Term, 


Mill,  Mr., 
Mind, 


Reasoning  conforms  to  logical  rules,  Section  73. 

"  every  trutli  established  by,  is  developed 

by  a  process  of  Arithmetic,  Geometry,  or 
Analj'sis,  or  a  combination  of  them,  96. 
.The  science  of  quantity,  76. 

Pare,  embraces  the  principles  of  the  science,  98  to  104. 
"      on  what  based,  100. 

Mixed,  embraces  the  applications,  104. 

Primary  signification,  99. 

Language  of,  90. 

'•'  Exact  science,"  100. 

Logical  test  of  truth  in,  100. 

a  deductive  science,  100,  101. 

concerned  with  number  and  space,  73,  76,  108. 

What  gives  rise  to  its  existence,  103. 

Why  peculiarly  adapted  to  give  clear  ideas,  374^376, 
379. 

a  pure  science,  379. 

considered  as  furnishing  the  keys  of  knowledge,  381. 

Widest  applications  are  in  nature,  384. 

Eifects  on  the  mind  and  character,  378,  390. 

Guidance  through  Nature,  390. 

Its  necessity  in  Astronomy,  391. 

Results  reached  by  it,  399,  400. 

Practical  advantages  of,  405. 

What  a  course  of,  should  present,  and  how,  366. 

Reasonings  of,  the  same  in  each  branch,  349. 

Faculties  required  by,  351. 

Necessity  of,  to  the  philosopher,  page  16. 

.A  term  of -comparison,  105. 

Unit  of,  should  be  exhibited  to  give  ideas  of  num- 
bers, 140. 
"        for  lines,  surfaces,  volumes,  253. 

of  a  magnitude,  how  ascertained,  253. 

distributed  when  the  predicate  of  a  negative  proposi- 
tion, 64. 

When  equivocal,  67. 

his  views  of  Logic,  72,  74. 

Operations  of,  in  reasoning,  6. 

Abstraction  a  faculty,  process,  and  state  of,  13. 

Processes  of,  which  leave  no  trace,  68. 
18 


410 


INDEX. 


Mind, 


Minus  sign. 
Motion    ■ 
Multiijlication, 


Multiplication, 


Faculties  of,  cultivated  by  Aritlimetic,  Section  188. 
Thinking  faculty  of,  peculiarly  cultivated  by  mathe- 
matics, 375,  376. 
Power  of,  fixed  by  definition,  301. 
proportional  to  force  impressed,  272. 
Readings  in,  130  ;  examples  in,  161. 
What  the  derinition  of,  requires,  185. 
Combinations  in,  199. 

All  operations  in,  governed  by  one  principle,  286. 
in  Algebra,  illustrations  of,  303-305. 


Names,  Definitions  are  of,  1. 

"  given  to  portions  of  space,  and  defined  in  Geometry, 

242. 
Naturalist  determines  the  species  of  an  animal  from  examining 

a  bone,  383. 
Negative  premises,  nothing  can  be  inferred  from,  67. 

"  demonstration,    its  nature,  267-369  ;   illustration  of 

268. 
Newton,  his  method  of  discovery,  32. 

"  changed  Astronomy  from  an  experimental  to  a  de- 

ductive science,  387-389. 
Lemmas,  329-337. 

"  in  harmony  with  Leibnitz,  341. 

Non-distribution  of  terms,  61. 

"  Word  "  some  "  which  marks,  not  always  expressed,  62. 

Number A  unit,  or  a  collection  of  units,  78. 

"  How  learned,  78. 

"  Axioms  for  forming,  78,  308. 

"  Three  ways  of  expressing,  113. 

"  Ideas  of,  complex,  115. 

"  Two  things  necessary  for  apprehending  clearly,  117. 

"  Simple  and  Denominate,  118. 

"  Examples  of  reading  Simple,  137. 

"  Two  ways  of  forming  from  ONE,  138. 

"  first  learned  through  the  senses,  140,  366. 

"  Two  ways  of  comparing,  171. 

"  compared,  must  be  of  the  same  kind,  179-183. 

"  Definitions  of,  205,  206. 

**  must  be  of  something,  279. 

**  may  stand  for  all  things,  280. 


INDEX. 


411 


Number First  lessons  in,  impress  tlie  first  elements  of  mathe- 
matical science.  Section  352. 


Olmsted's  Mechanics,  quotation  from,  273. 

Optician,  Illustration,  216. 

Oral  Arithmetic,  its  inefficiency  without  figures,  223. 
Order  of  subjects  in  Arithmetic,  190. 

Parallelogram... A  quadrilateral  having  its  opposite  sides,  taken  two 
and,  two  parallel,  246. 
"  regarded  as  a  species,  17 ;  as  a  genus,  18. 

"  Properties  of,  260. 

Particular  proposition,  62. 

"  premises,  nothing  can  be  proved  from,  67. 

Pendulum,  the  standard  for  measurement,  257. 

Philosophy,  Natural,  originally  experimental,  387. 

"  "         has  been  rendered  mathematical,  387. 

Place,  idea  attached  to  the  word,  81. 

"  designates  the  unit  of  a  number,  206. 

Plane That  with  which  a  straight  line,  having  two  points  in 

common,  and  anyhow  placed,  will  coincide,  244. 
"  First  idea  of,  how  impressed,  369. 

Plane  Figure.  .Any  portion  of  a  plane  bounded  by  lines,  244. 
Plane  Figures       in  general,  247. 

Point That  which  has  position  in  space  without  occupying 

any  part  of  it,  80. 
Points,  extremities  or  limits  of  a  line,  243. 

Practical  Rules  in  Arithmetic,  185,  186. 

"  The  true,  211,  must  be  the  consequent  of  science,  232. 

"  •        Popular  meaning  of,  401,  403. 

"  Questions  with  regard  to,  401,  402. 

"  Consequences  of  an  erroneous  view  of,  404. 

"  True  signification  of,  404. 

Practice  precedes  theory,  but  is  improved  by  it,  42. 

"  without  science  is  empiricism,  page  13. 

Predicate That  which  is  affirmed  or  denied  of  the  subject,  38; 

"  Distribution,  03. 

"  Non-distribution,  63. 

"  sometimes  coincides  with  the  subject,  63. 

Premise Each  of  two  propositions  of  a  syllogism  admitted  to 

be  true,  40. 


413 


IKDEX. 


Premise 


Pressure, 
Principle 


Principles 


Process 

Product 

Progression, 

Property 

Proportion 


Proposition 


..Major  Premise — The  proposition  of  a  syllogism 
which  contains  the  predicate  of  the  conclusion.  Sec- 
tion 40. 

Minor  Premise — The   proposition  of  a  syllogism 
which  contains  the  subject  of  the  conclusion,  40. 

a  law  of  fluids,  414. 

of  science  applied,  22. 

on  which  valid  arguments  are  constructed,  52. 

Value  of  a,  greater  as  it  is  more  simple,  54. 

Aristotle's  Dictum,  a  general,  55. 

the  same  in  the  ground  rules  for  simple  and  denomi- 
nate numbers,  159-162,  236. 

of  science  and  rule  of  art,  187. 

should  be  separated  from  apjjlications,  194, 195. 

of  science  are  general  truths,  212. 

of  Arithmetic,  how  taught,  212. 

should  precede  practice,  233. 

of  Mathematics,  deduced  from  definitions  and  axioms, 
301. 

of  acquiring  mathematical  knowledge,  366-370. 

of  several  numbers,  298. 

Geometrical,  178. 

of  a  figure,  260. 

.  The  relation  which  one  quantity  bears  to  another  with 
respect  to  its  being  greater  or  less,  171,  271-273. 

Arithmetical  and  Geometrical,  171. 

Reciprocal  or  Inverse,  273. 

of  geometrical  figures,  274r-277. 
.A  judgment  expressed  in  words,  35. 

All  truth  and  all  error  lie  in  propositions,  also  answers 
to  all  questions,  36. 

formed  by  putting  together  two  names,  37. 

consists  of  three  parts,  38. 

subject  and  predicate,  called  extremes,  38. 

Affirmative,  39  ;  Negative,  39. 

Three  propositions  essential  to  a  syllogism,  40. 

Universal,  62. 

Particular,  62. 


Quadrilateral.. A  portion  of  a  plane  bounded  by  four  straight  lines, 
246. 


INDEX. 


413 


Quadrilateral  regarded  as  a  genus,  Section  17.  ^ 

"  Different  varieties  of,  246. 

Quality  of  a  proposition  refers   to  its    being  affirmative  or 

negative,  63. 
Quantities  only  of  the  same  kind  can  be  compared,  271. 

"  Two  classes  of,  in  Algebra,  293,  317. 

"  "         "        "     in  the  other  branches   of   Analysis, 

286,  287,  317. 
"  compared,  must  be  equal  or  unequal,  109,  311. 

Quantity Is  a  general  terra  applicable  to  every  thing  which  can 

be  increased  or  diminished,  and  measured,  75, 371. 
Abstract,  75,  107. 
"  Concrete,  107. 

,"  Propositions  divided  according  to,  62. 

"  presented  by  symbols,  89. 

"  consists  of  parts  which  can  be  numbered,  380. 

"  Constant,  286. 

"  Variable,  286. 

"  Sis  operations  can  be  performed  on,  292, 299. 

"  represented  by  six  signs,  293. 

"  Nature  of,  not  affected  by  the  sign,  294,  300. 

Questiong  known,  when  all  propositions  are  known,  36. 

Analysis  of,  183,  184. 
"  with  regard  to  methods  of  instruction,  353. 

Quotations  from  Kant,  21 ;  Sir  John  Herschel,  27,  372,  391,  409 ; 

Cousin,  188 ;  Olmsted's  Mechanics,  272 ;  Locke, 
870;  Mansfield's  Discourse  on  Mathematics,  375, 
377 ;  Lord  Bacon,  378  ;  Dr.  Barrow,  378,  390. 


Railways,  Problem  presented  in,  411. 

Rainbow,  Illustration,  372. 

Ratio The  quotient  arising   from  dividing  one  number  or 

quantity  by  another,  171,  271. 

"  Discussion  concerning  it,  173-179. 

"  Arithmetical  and  Geometrical,  171. 

•'  How  determined,  173. 

"  .  An  abstract  number,  271,  276. 

"  Terms  direct,  inverse,  or  reciprocal,  not  applicable  to, 

273. 
Reading  in  Addition,  123,  124 ;  advantages  of,  135. 

"  in  Subtraction,  127. 


414 


IISTDEX. 


Readme 


Reason, 


in  Multiplication,  Section  129. 
in  Division,  130, 

of  figures,  its  aid  in  practical  operations,  234. 
To  make  use  of  arguments,  42. 
"  A  premise  placed  after  the  conclusion,  48. 

Reasoning The  act  of  proceeding  from   certain  judgments  to 

another,  founded  on  them,  9. 
"  Three  operations  of  the  mind  concerned  in,  6. 

"  Process,  sameness  of  the,  42,  43,  45,  318. 

"  processes  of  mathematics  consist  of  two  parts,  73. 

"  in  Analysis  is  based  on  the  supposition  that  we  are 

dealing  with  things,  282. 
Reciprocal  or        Inverse  Proportion,  270. 

Rectangle A  parallelogram  whose  angles  are  right  angles,  246. 

Remarks,  Concluding  subject  of  Arithmetic,  240. 

Reservoirs,  Croton,  description  of,  412. 
Right  angle,  Definition  of,  262. 

Roman  Table,       when  taught,  219. 

Symbol  for  the  extraction  of,  299« 

Solution  of  questions  in,  177. 

Comparison  of  numbers,  194. 

should  precede  its  applications,  195. 

Every  thing  done  according  to,  21. 

of  reasoning  analogous  to  those  of  Arithmetic,  45. 

Advantages  of  logical,  50. 

for  teaching,  194. 

How  framed,  301. 


Root, 

Rule  of  Three, 


Rules, 


Scale  of  Tens,       Units  increasing  by,  131-137,  165,  191. 

Science In  its  popular  sense  means  knowledge   reduced  to 

order,  21,  276. 
"  In  its  technical  sense  means  an  analysis  of  the  laws 

of  nature,  21. 
contrasted  with  art,  22, 
of  Arithmetic,  180. 
Principles  of,  204,  212. 

•Methods  of,  must  be  followed  in  Arithmetic,  232. 
of  Geometry,  241,  252,  261. 
Objects  and  means  of  pure,  372. 
should  be  made  as  much  deductive  as  possible,  386. 
Deductive  and  experimental,  387. 


IKDEX.  415 


Science wlien  experimental,  Sections  388,  389  ;  Avlien  deduc- 
tive, 388,  389. 
■  "  What  it  has  accomplished,  398. 

"  Practical  value  of,  in  factories,  408. 

"  "  "        "    in  constructing  steamships,  409. 

"  "  "        "    in  laying  out  and  measuring  land, 

410. 
"  "  •  "        "    in  constructing  railways,  411. 

"  Its  power  illustrated  in  Croton  aqueduct,  413. 

"  What  constitutes  it,  354. 

Second  Arithmetic,  its  place  and  construction,  231-234. 
Sextant,  its  uses  in  Navigation,  409. 

Shades,  Shadows,  and  Perspective — An  application  of  Descriptive 

Geometry,  363. 
SiGNiFiCATE  . . .  .An  individual  for  which  a  common  term  stands,  15. 
Signs,  Six  used  to  denote  operations  on  quantity,  293. 

"  ■  How  to  be  interpreted,  294. 

"  do  not  affect  the  nature  of  the  quantity,  294,  300. 

"  indicate  operations,  300. 

Solution  of  all  questions  in  the  Rule  of  Three,  177. 

"  of  an  equation  in  Algebra,  312. 

Space Is  indefinite  extension,  80. 

"  has  three  dimensions,  length,  breadth,  and  thickness, 

80-82. 
"                        Clear  conception  of,  necessary  to  understand  Geome- 
try, 242. 
Species One  of  the  divisions  of  a  genus  in  which  the  charac- 
teristic is  less  extensive,  but  more  full  and  com- 
plete, 16,  17. 
Subspecies — One  of  the  divisions   of  a   species,  in 
wliicli  the  characteristic  is  less  extensive,  but  more 
full  and  complete,  16,  19. 
Lowest  Species — A  species  which  cannot  be  regard- 
ed as  a  genus,  17. 
Spelling,  120 ;  in  Addition,  &c.,  122-130. 

Square A  quadrilateral  whose   sides  are   equal   and  angles 

right  angles,  249. 
Statement  of  a  proposition  in  Algebra,  312. 

"  in  what  it  consists,  313. 

Steamship,  an  application  of  science,  409. 


416  IN^DEX, 


Subject The   name   denoting  tlie  person  or  tMng  of  which 

something  is  affirmed  or  denied,  Section  38. 
Subjects,  How  presented  in  a  text-book,  213-216. 

Subtraction,  Headings  in,  127. 

"  Examples  in,  160. 

"  Combinations  in,  198. 

"  All  operations  in,  governed  by  one  principle,  236. 

"  in  Algebra,  illustration  o3, 302. 

Suggestions,         for  teaching  G-eometry,  277. 

"  for  teaching  Algebra,  319. 

Sum,  Its  definition,  207. 

Surface A  portion  of  space  having  two  dimensions,  83,  244, 

369. 
"  Plane  and  Curved,  83, 244. 

Surfaces,  Curved,  249. 

"  "      of  Elementary  Geometry,  249. 

"  Limits  of,  251. 

SuKVETiNG The  application  of  the  principles  of  Trigonometry  to 

the  measurement  of  portions  of  the  earth's  surface, 
361. 
"  A  branch  of  practical  science,  410. 

Syllogism.  ....  .A  form  of  stating  the  connection  which  may  exist  for 

the  purpose  of  reasoning,  between  three  proposi- 
tions, 40. 
"  A  formula  for  ascertaining  what  may  be  predicated. 

How  it  accomplishes  this,  41. 
"  not  meant  by  Aristotle  to  be  the  form  in  which  argu- 

ments should  always  be  stated,  53. 
"  not  a  distinct  kind  of  argument,  54. 

*'  an  argument  stated  at  full  length,  56. 

"  Symbols  used  for  the  terms  of,  56. 

"  Rules  for  examining  syllogisms,  67. 

"  has  three  and  only  three  terms,  67. 

"  "        "      "        "        "     propositions,  67. 

"  test  of  deductive  reasoning,  72, 102,  311. 

Symbols The  letters  which  denote  quantities,  and  the  signs 

which  indicate  operations,  93,  89,  300. 
"  used  for  the  tei-ms  of  a  syllogism,  58. 

"  Advantages  of,  57. 

•*  Validity  of  the  argument  still  evident,  58. 

**  Truths  inferred  by  means  of,  true  of  all  things,  381. 


INDEX.  417 


Symbols regarded  as  things.  Section  283. 

"  Two  classes  of,  in  analysis,  310. 

"  Abstract  and  concrete  quantity  represented  by,  371. 

Synthesis The  process  of  first  considering  the  elements  sepa^ 

rately,  then  combining  them,  and  ascertaining  the 
results  of  combination,  95,  377. 
Synthetical  form,  for  what  best  adapted,  71,  95. 

Tables  of  Denominate  Numbers,  141-158. 

Tangent Tangent  and  Limit,  327,  328. 

Technical Particular  and  limited  sense,  90. 

Term Is  an  act  of  apprehension  expressed  in  words,  15. 

"  A  singular  term  denotes  but  a  single  individual,  15. 

"  A  common  denotes  any  individual  of  a  Avliole  class,  15. 

"  "         affords  the  means  of  classification,  16. 

"  "         Nature  of,  20. 

"  "         No  real  thing  corresponding  to,  20. 

**  "         Why  applicable  to  several  individuals,  20. 

Major  Term — The  predicate  of  the  conclusion,  40. 
Minor  Term — The  subject  of  the  conclusion,  40. 
Middle    Term — The    common    term  of    the    two 
premises,  40. 
"  Distributed— A  term  is  distributed  when  it  stands 

for  all  its  significates,  61. 
"  NOT  distributed — When  it  stands  for  a  part  of  its 

significates  only,  61. 

Terms Two  of  the  three  parts  of  a  proposition,  38. 

"  The  antecedent  and  consequent  of  a  proportion,  173. 

"  should  always  be  used  in  the  same  sense,  178,  209. 

Text-Book Should  be  an  aid  to  the  teacher  in  imparting  instruc- 
tion, and  to  the  learner  in  acquiring  knowledge, 
213. 

Thickness A  dimension  of  space,  81. 

Third  Arithmetic,  Principles  contained  in,  and  method  of  construction 

235-240. 
Time,  Measure,     its  units  and  scale,  155. 
Topograpny,         Its  uses,  410. 

Trapezoid A  quadrilateral,  having  two  sides  parallel,  246. 

Triangle A  portion  of  a  plane  bounded  by  three  straight  lines 

345. 

18* 


418  IKDEX. 


Triangle The  simplest  plane  figure.  Section  245. 

"  Different  kinds  of,  245. 

"  regarded  as  a  genus,  260. 

Trigonometry  .  .An  application  of  the  principles  of  Arithmetic,  Alge- 
bra, and  Geometry  to   the   determination   of  the 
sides  and  angles  of  triangles,  360. 
Plane  and  Spherical,  360. 
Troy  Weight,       Its  units  and  scale,  144. 

Truth An  exact  accordance  with  what  has  been,  is,  or  shall 

be,  24. 
"  Two  methods  of  ascertaining,  24. 

"  is  inference  from  facts  or  other  truths,  24,  35. 

"  regarded  as  a  species.  25. 

"  How  inferred  from  facts,  26. 

"  A  true  proposition.  36. 

I'ruths Intuitive  or  Self-evident — Are  such  as  becoma 

known  by   considering    all    the   facts    on    which 
they  depend,  and    apprehending  the  relations  of 
those  facts  at  the  same  time,  and  by  the  same  act 
by  which  we  apprehend  the  facts  themselves,  27. 
**  Logical — Those  inferred  from  numerous  and  com- 

plicated  facts ;     and    also,  truths    inferred  from 
truths,  28. 
*•  of  Geometry,  241. 

'*  Three  classes  of,  241. 

"  Demonstrative,  241. 

Unit  fixed  by  the  place  of  the  figure,  134. 

"  of  the  fraction,  168, 169. 

"  of  the  expression,  168. 

Unities,  Advantages  of  the  system  of,  157-162. 

Unit  of  MEASURE...The  standard  for  measurement,  105. 
"  for  lines,  surfaces,  volumes,  253. 

"  only  basis  for  estimating  quantity,  255. 

Unit  one A  single  thing.  111. 

All  numbers  come  from,  115,  116,  139, 157. 
"  Method  of  impressing  its  values,  140. 

"  Three  kinds  of  operations  performed  upon,  190-194. 

Units,  Abstract  or  simple,  118, 139. 

**  Denominate  or  Concrete,  118, 

*  of  currency,  139. 


IKDEX. 


419 


Units  of  weight,  Section  139. 

"  of  measure,  139, 146, 253. 

••  of  length,  147. 

"  of  surface,  148. 

•'  Duodecimal,  149. 

"  of  volume,  153. 

"  Fractional,  163, 193, 

United  States,  Currency,  141. 

Unity — Unit  . . .  Any  thing  regarded  as  a  whole,  116, 117. 

Universal  Proposition,  63. 

Utility  and  Progress,  leading  ideas,  page  11. 

Vabiables Quantities  which  undergo  certain  changes  of  value, 

the  laws  of  which  are  indicated  by  the  algebraic 
expressions  into  which  they  enter,  286,  387,  363. 
"  represented  by  the  final  letters  of  the  alphabet,  288. 

Variations,  Theory  of,  389. 

Varying  Scales,    Units  increasing  by,  138, 191. 
Velocity  known  by  measurement,  95. 

Volume A  portion  of,  space  having  three  dimensions,  84. 

"  A  portion  of  space  combining  the  three  dimensions 

of  length,  breadth,  and  thickness,  350,  370. 
Limit  of,  297. 
"  First  idea  of,  how  impressed,  370. 

Volumes bounded  by  plane  and  curved  surfaces,  84 

"  Three  classes  of,  350. 

"  Analysis  of,  comparison,  375,  376. 

"  Comparison  of,  under  the  supposition  of  changes,  376. 


Weight 


Whately, 
Words, 


known  by  measurement,  106. 

should  be  exhibited  to  give  ideas  of  numbers,  140. 

Standard  for,  258. 

Archbishop,  his  views  of  logic,  72. 

Definition  of,  120. 

expressing  results  of  combmations,  197,  201. 

Double,  or  incomplete  sense  of,  372. 


Zebo. 


.The  limit  of  a  decreasing  quantity,  306-310. 
what  quantities  are  denoted  by  it,  343. 


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